import { type PracticeQuestion } from "../../types/lesson";

export const RIGHT_TRI_TRIG_EASY: PracticeQuestion[] = [
  {
    id: "64c1f044",
    type: "mcq",
    questionHtml:
      "One angle is a right angle.<br>The measure of a second angle is x°.<br>The length of the leg opposite the angle with measure x° is 26.<br>The length of the leg adjacent to the angle with measure x° is 7.<br>A note indicates the figure is not drawn to scale.<br><br>In the triangle shown, what is the value of <strong>tan x °</strong>?",
    choices: [
      { label: "A", text: "<strong>one twenty sixth</strong>" },
      { label: "B", text: "<strong>(19) / (26)</strong>" },
      { label: "C", text: "<strong>(26) / (7)</strong>" },
      { label: "D", text: "<strong>(33) / (7)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The tangent of an acute angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the shorter side adjacent to the angle. In the triangle shown, the length of the side opposite the angle with measure <strong>x °</strong> is <strong>26</strong> units and the length of the side adjacent to the angle with measure <strong>x °</strong> is <strong>7</strong> units. Therefore, the value of <strong>tan x °</strong> is <strong>(26) / (7)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/64c1f044_svg1.svg",
  },
  {
    id: "b0c5ece5",
    type: "mcq",
    questionHtml:
      "One angle is a right angle.<br>The length of the side opposite the right angle is 19.<br>The lengths of the other 2 sides are as follows:<br><br>b<br>4<br><br>A note indicates the figure is not drawn to scale.<br><br>Which equation shows the relationship between the side lengths of the given triangle?",
    choices: [
      { label: "A", text: "<strong>4 b = 19</strong>" },
      { label: "B", text: "<strong>4 + b = 19</strong>" },
      { label: "C", text: "<strong>4² + b² = 19²</strong>" },
      { label: "D", text: "<strong>4² − b² = 19²</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. Therefore, <strong>a² + b² = c²</strong>, where <strong>a</strong> and <strong>b</strong> are the lengths of the legs and <strong>c</strong> is the length of the hypotenuse. For the given right triangle, the lengths of the legs are <strong>4</strong> and <strong>b</strong>, and the length of the hypotenuse is <strong>19</strong>. Substituting <strong>4</strong> for <strong>a</strong> and <strong>19</strong> for <strong>c</strong> in the equation <strong>a² + b² = c²</strong> yields <strong>4² + b² = 19²</strong>. Thus, the relationship between the side lengths of the given triangle is <strong>4² + b² = 19²</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/b0c5ece5_svg1.svg",
  },
  {
    id: "c9f8d1e9",
    type: "mcq",
    questionHtml:
      "The side lengths are labeled as follows:<br><br>a<br>b<br>c<br><br>The angle opposite the side labeled c is a right angle.<br>A note indicates the figure is not drawn to scale.<br><br>For the right triangle shown, <strong>a = 4</strong> and <strong>b = 5</strong>. Which expression represents the value of <strong>c</strong>?",
    choices: [
      { label: "A", text: "<strong>4 + 5</strong>" },
      { label: "B", text: "<strong>√((4) (5))</strong>" },
      { label: "C", text: "<strong>√(4 + 5)</strong>" },
      { label: "D", text: "<strong>√(4² + 5²)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. By the Pythagorean theorem, if a right triangle has a hypotenuse with length <strong>c</strong> and legs with lengths <strong>a</strong> and <strong>b</strong>, then <strong>c² = a² + b²</strong>. In the right triangle shown, the hypotenuse has length <strong>c</strong> and the legs have lengths <strong>a</strong> and <strong>b</strong>. It's given that <strong>a = 4</strong> and <strong>b = 5</strong>. Substituting <strong>4</strong> for <strong>a</strong> and <strong>5</strong> for <strong>b</strong> in the Pythagorean theorem yields <strong>c² = 4² + 5²</strong>. Taking the square root of both sides of this equation yields <strong>c = + or − √(4² + 5²)</strong>. Since the length of a side of a triangle must be positive, the value of <strong>c</strong> is <strong>√(4² + 5²)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/c9f8d1e9_svg1.svg",
  },
];

export const RIGHT_TRI_TRIG_MEDIUM: PracticeQuestion[] = [
  {
    id: "13d9a1c3",
    type: "spr",
    questionHtml:
      "In the right triangle shown above, what is the length of <strong>line P Q</strong> ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 4. Triangle PQR has given angle measures of 30° and 90°, so the third angle must be 60° because the measures of the angles of a triangle sum to 180°. For any special right triangle with angles measuring 30°, 60°, and 90°, the length of the hypotenuse (the side opposite the right angle) is 2x, where x is the length of the side opposite the 30° angle. Segment PQ is opposite the 30° angle. Therefore, 2(PQ) = 8 and PQ = 4.",
    hasFigure: true,
    figureUrl: "/practice-images/13d9a1c3_img1.png",
  },
  {
    id: "33e29881",
    type: "mcq",
    questionHtml:
      "In right triangle <strong>R S T</strong>, the sum of the measures of angle <strong>R</strong> and angle <strong>S</strong> is <strong>90</strong> degrees. The value of <strong>sin (R)</strong> is <strong>(√(15)) / (4)</strong>. What is the value of <strong>cos (S)</strong>?",
    choices: [
      { label: "A", text: "<strong>(√(15)) / (15)</strong>" },
      { label: "B", text: "<strong>(√(15)) / (4)</strong>" },
      { label: "C", text: "<strong>(4 √(15)) / (15)</strong>" },
      { label: "D", text: "<strong>√(15)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The sine of any acute angle is equal to the cosine of its complement. It’s given that in right triangle <strong>R S T</strong>, the sum of the measures of angle <strong>R</strong> and angle <strong>S</strong> is <strong>90</strong> degrees. Therefore, angle <strong>R</strong> and angle <strong>S</strong> are complementary, and the value of <strong>sin R</strong> is equal to the value of <strong>cos S</strong>. It's given that the value of <strong>sin R</strong> is <strong>(√(15)) / (4)</strong>, so the value of <strong>cos S</strong> is also <strong>(√(15)) / (4)</strong>.<br>Choice A is incorrect. This is the value of <strong>tan S</strong>.<br>Choice C is incorrect. This is the value of <strong>(1) / (cos S)</strong>.<br>Choice D is incorrect. This is the value of <strong>(1) / (tan S)</strong>.",
    hasFigure: false,
  },
  {
    id: "902dc959",
    type: "mcq",
    questionHtml:
      "In the figure above, what is the value of <strong>tan A</strong>?",
    choices: [
      { label: "A", text: "<strong>20 over 29</strong>" },
      { label: "B", text: "<strong>21 over 29</strong>" },
      { label: "C", text: "<strong>20 over 21</strong>" },
      { label: "D", text: "<strong>21 over 20</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Angle A is an acute angle in a right triangle, so the value of tan(A) is equivalent to the ratio of the length of the side opposite angle A, 20, to the length of the nonhypotenuse side adjacent to angle A, 21. Therefore,  <strong>tan A = 20 over 21</strong>.Choice A is incorrect. This is the value of sin(A). Choice B is incorrect. This is the value of cos(A). Choice D is incorrect. This is the value of tan(B).",
    hasFigure: true,
    figureUrl: "/practice-images/902dc959_img1.png",
  },
  {
    id: "9ec76b54",
    type: "mcq",
    questionHtml:
      "A right triangle has legs with lengths of <strong>28</strong> centimeters and <strong>20</strong> centimeters. What is the length of this triangle's hypotenuse, in centimeters?",
    choices: [
      { label: "A", text: "<strong>8 √(6)</strong>" },
      { label: "B", text: "<strong>4 √(74)</strong>" },
      { label: "C", text: "<strong>48</strong>" },
      { label: "D", text: "<strong>1, 184</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. It's given that the right triangle has legs with lengths of <strong>28</strong> centimeters and <strong>20</strong> centimeters. Let <strong>c</strong> represent the length of this triangle's hypotenuse, in centimeters. Therefore, by the Pythagorean theorem, <strong>28² + 20² = c²</strong>, or <strong>1, 184 = c²</strong>. Taking the positive square root of both sides of this equation yields <strong>√(1, 184) = c</strong>, or <strong>4 √(74) = c</strong>. Therefore, the length of this triangle's hypotenuse, in centimeters, is <strong>4 √(74)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect. This is the square of the length of the triangle’s hypotenuse.",
    hasFigure: false,
  },
  {
    id: "a5aee181",
    type: "mcq",
    questionHtml:
      "The length of a rectangle’s diagonal is <em>(expression)</em>, and the length of the rectangle’s shorter side is <em>(expression)</em>. What is the length of the rectangle’s longer side?",
    choices: [
      { label: "A", text: "<em>(expression)</em>" },
      { label: "B", text: "<em>(expression)</em>" },
      { label: "C", text: "<em>(expression)</em>" },
      { label: "D", text: "<em>(expression)</em>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. A rectangle’s diagonal divides a rectangle into two congruent right triangles, where the diagonal is the hypotenuse of both triangles. It’s given that the length of the diagonal is <strong>5 √(17)</strong> and the length of the rectangle’s shorter side is <strong>5</strong>. Therefore, each of the two right triangles formed by the rectangle’s diagonal has a hypotenuse with length <strong>5 √(17)</strong>, and a shorter leg with length <strong>5</strong>. To calculate the length of the longer leg of each right triangle, the Pythagorean theorem, <strong>a² + b² = c²</strong>, can be used, where <strong>a</strong> and <strong>b</strong> are the lengths of the legs and <strong>c</strong> is the length of the hypotenuse of the triangle. Substituting <strong>5</strong> for <strong>a</strong> and <strong>5 √(17)</strong> for <strong>c</strong> in the equation <strong>a² + b² = c²</strong>  yields <strong>5² + b² = (5 √(17))²</strong>, which is equivalent to <strong>25 + b² = 25 (17)</strong>, or <strong>25 + b² = 425</strong>. Subtracting <strong>25</strong> from each side of this equation yields <strong>b² = 400</strong>. Taking the positive square root of each side of this equation yields <strong>b = 20</strong>. Therefore, the length of the longer leg of each right triangle formed by the diagonal of the rectangle is <strong>20</strong>. It follows that the length of the rectangle’s longer side is <strong>20</strong>.<br>Choice A is incorrect and may result from dividing the length of the rectangle’s diagonal by the length of the rectangle’s shorter side, rather than substituting these values into the Pythagorean theorem.<br>Choice C is incorrect and may result from using the length of the rectangle’s diagonal as the length of a leg of the right triangle, rather than the length of the hypotenuse.<br>Choice D is incorrect. This is the square of the length of the rectangle’s longer side.",
    hasFigure: false,
  },
  {
    id: "bcb66188",
    type: "mcq",
    questionHtml:
      "Triangle <strong>F G H</strong> is similar to triangle <strong>J K L</strong>, where angle <strong>F</strong> corresponds to angle <strong>J</strong> and angles <strong>G</strong> and <strong>K</strong> are right angles. If <strong>sin (F) = (308) / (317)</strong>, what is the value of <strong>sin (J)</strong>?",
    choices: [
      { label: "A", text: "<strong>(75) / (317)</strong>" },
      { label: "B", text: "<strong>(308) / (317)</strong>" },
      { label: "C", text: "<strong>(317) / (308)</strong>" },
      { label: "D", text: "<strong>(317) / (75)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. If two triangles are similar, then their corresponding angles are congruent. It's given that right triangle <strong>F G H</strong> is similar to right triangle <strong>J K L</strong> and angle <strong>F</strong> corresponds to angle <strong>J</strong>. It follows that angle <strong>F</strong> is congruent to angle <strong>J</strong> and, therefore, the measure of angle <strong>F</strong> is equal to the measure of angle <strong>J</strong>. The sine ratios of angles of equal measure are equal. Since the measure of angle <strong>F</strong> is equal to the measure of angle <strong>J</strong>, <strong>sin (F) = sin (J)</strong>. It's given that <strong>sin (F) = (308) / (317)</strong>. Therefore,  <strong>sin (J)</strong> is <strong>(308) / (317)</strong>.<br>Choice A is incorrect. This is the value of <strong>cos (J)</strong>, not the value of <strong>sin (J)</strong>.<br>Choice C is incorrect. This is the reciprocal of the value of <strong>sin (J)</strong>, not the value of <strong>sin (J)</strong>.<br>Choice D is incorrect. This is the reciprocal of the value of <strong>cos (J)</strong>, not the value of <strong>sin (J)</strong>.",
    hasFigure: false,
  },
  {
    id: "bf8d843e",
    type: "mcq",
    questionHtml:
      "In <strong>triangle A, B C</strong> above, what is the length of  <strong>A, D</strong>?",
    choices: [
      { label: "A", text: "4" },
      { label: "B", text: "6" },
      { label: "C", text: "<strong>6 · the √ 2</strong>" },
      { label: "D", text: "<strong>6 · the √ 3</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Triangles ADB and CDB are both <strong>30 ° 60 ° 90 °</strong> triangles and share <strong>side B D</strong> . Therefore, triangles ADB and CDB are congruent by the angle-side-angle postulate. Using the properties of <strong>30 ° 60 ° 90 °</strong> triangles, the length of <strong>side A, D</strong> is half the length of hypotenuse <strong>A, B</strong> . Since the triangles are congruent, <strong>the length of side A, B = the length of side B C, which = 12</strong>. So the length of <strong>side A, D</strong> is <strong>twelve halves = 6</strong>.Alternate approach: Since angle CBD has a measure of <strong>30 °</strong>, angle ABC must have a measure of <strong>60 °</strong>. It follows that triangle ABC is equilateral, so side AC also has length 12. It also follows that the altitude BD is also a median, and therefore the length of AD is half of the length of AC, which is 6.<br>Choice A is incorrect. If the length of <strong>side A, D</strong> were 4, then the length of <strong>side A, B</strong> would be 8. However, this is incorrect because <strong>side A, B</strong> is congruent to <strong>side B C</strong>, which has a length of 12. Choices C and D are also incorrect. Following the same procedures as used to test choice A gives <strong>side A, B</strong> a length of <strong>12 · the √ 2</strong> for choice C and <strong>12 · the √ 3</strong> for choice D. However, these results cannot be true because <strong>side A, B</strong> is congruent to <strong>side B C</strong>, which has a length of 12.",
    hasFigure: true,
    figureUrl: "/practice-images/bf8d843e_img1.png",
  },
  {
    id: "de550be0",
    type: "mcq",
    questionHtml:
      "One angle is a right angle.<br>The length of the side opposite the right angle is 21.<br>The length of one side adjacent to the right angle is 6.<br>The length of the other side adjacent to the right angle is a.<br>A note indicates the figure is not drawn to scale.<br><br>For the triangle shown, which expression represents the value of <strong>a</strong>?",
    choices: [
      { label: "A", text: "<strong>√(21² − 6²)</strong>" },
      { label: "B", text: "<strong>21² − 6²</strong>" },
      { label: "C", text: "<strong>√(21 − 6)</strong>" },
      { label: "D", text: "<strong>21 − 6</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. For the right triangle shown, the lengths of the legs are <strong>a</strong> units and <strong>6</strong> units, and the length of the hypotenuse is <strong>21</strong> units. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. Therefore, <strong>a² + 6² = 21²</strong>. Subtracting <strong>6²</strong> from both sides of this equation yields <strong>a² = 21² − 6²</strong>. Taking the square root of both sides of this equation yields <strong>a = + or − √(21² − 6²)</strong>. Since <strong>a</strong> is a length, <strong>a</strong> must be positive. Therefore, <strong>a = √(21² − 6²)</strong>. Thus, for the triangle shown, <strong>√(21² − 6²)</strong> represents the value of <strong>a</strong>.<br>Choice B is incorrect. For the triangle shown, this expression represents the value of <strong>a²</strong>, not <strong>a</strong>.<br>Choice C is incorrect and may result from conceptual errors.<br>Choice D is incorrect and may result from conceptual errors.",
    hasFigure: true,
    figureUrl: "/practice-images/de550be0_svg1.svg",
  },
];

export const RIGHT_TRI_TRIG_HARD: PracticeQuestion[] = [
  {
    id: "0e709a29",
    type: "mcq",
    questionHtml:
      "<strong>R S = 440</strong><br><strong>S T = 384</strong><br><strong>T R = 584</strong><br>The side lengths of right triangle <strong>R S T</strong> are given. Triangle <strong>R S T</strong>  is similar to triangle <strong>U V W</strong>, where <strong>S</strong> corresponds to <strong>V</strong> and <strong>T</strong> corresponds to <strong>W</strong>. What is the value of <strong>tan W</strong>?",
    choices: [
      { label: "A", text: "<strong>(48) / (73)</strong>" },
      { label: "B", text: "<strong>(55) / (73)</strong>" },
      { label: "C", text: "<strong>(48) / (55)</strong>" },
      { label: "D", text: "<strong>(55) / (48)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The hypotenuse of triangle <strong>R S T</strong> is the longest side and is across from the right angle. The longest side length given is <strong>584</strong>, which is the length of side <strong>T R</strong>. Therefore, the hypotenuse of triangle <strong>R S T</strong> is side <strong>T R</strong>, so the right angle is angle <strong>S</strong>. The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side, which is the side across from the angle, to the length of the adjacent side, which is the side closest to the angle that is not the hypotenuse. It follows that the opposite side of angle <strong>T</strong> is side <strong>R S</strong> and the adjacent side of angle <strong>T</strong> is side <strong>S T</strong>. Therefore, <strong>tan T = (R S) / (S T)</strong>. Substituting <strong>440</strong> for <strong>R S</strong> and <strong>384</strong> for <strong>S T</strong> in this equation yields <strong>tan T = (440) / (384)</strong>. This is equivalent to <strong>tan T = (55) / (48)</strong>. It’s given that triangle <strong>R S T</strong> is similar to triangle <strong>U V W</strong>, where <strong>S</strong> corresponds to <strong>V</strong> and <strong>T</strong> corresponds to <strong>W</strong>. It follows that <strong>R</strong> corresponds to <strong>U</strong>. Therefore, the hypotenuse of triangle <strong>U V W</strong> is side <strong>W U</strong>, which means <strong>tan W = (U V) / (V W)</strong>. Since the lengths of corresponding sides of similar triangles are proportional, <strong>(R S) / (S T) = (U V) / (V W)</strong>. Therefore, <strong>tan W = (U V) / (V W)</strong> is equivalent to <strong>tan W = (R S) / (S T)</strong>, or <strong>tan W = tan T</strong>. Thus, <strong>tan W = (55) / (48)</strong>.<br>Choice A is incorrect. This is the value of <strong>cos W</strong>, not <strong>tan W</strong>.<br>Choice B is incorrect. This is the value of <strong>sin W</strong>, not <strong>tan W</strong>.<br>Choice C is incorrect. This is the value of <strong>(1) / (tan W)</strong>, not <strong>tan W</strong>.",
    hasFigure: false,
  },
  {
    id: "1429dcdf",
    type: "spr",
    questionHtml:
      "One angle is a right angle.<br>The measure of a second angle is x°.<br>The length of the hypotenuse is 23.<br>The length of the leg opposite the angle with measure x° is 16.<br>A note indicates the figure is not drawn to scale.<br><br>In the triangle shown, what is the value of <strong>sin x °</strong>?",
    choices: [],
    correctAnswer: ".6956, .6957, 16/23",
    explanation:
      "The correct answer is <strong>(16) / (23)</strong>. In a right triangle, the sine of an acute angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the triangle shown, the length of the side opposite the angle with measure <strong>x °</strong> is <strong>16</strong> units and the length of the hypotenuse is <strong>23</strong> units. Therefore, the value of <strong>sin x °</strong> is <strong>(16) / (23)</strong>. Note that 16/23, .6956, .6957, 0.695, and 0.696 are examples of ways to enter a correct answer.",
    hasFigure: true,
    figureUrl: "/practice-images/1429dcdf_svg1.svg",
  },
  {
    id: "25da87f8",
    type: "spr",
    questionHtml:
      "A triangle with angle measures 30°, 60°, and 90° has a perimeter of <strong>18 + 6 · the √ 3</strong>. What is the length of the longest side of the triangle?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 12. It is given that the triangle has angle measures of 30°, 60°, and 90°, and so the triangle is a special right triangle. The side measures of this type of special triangle are in the ratio <strong>2 to 1 to the √ 3</strong>. If x is the measure of the shortest leg, then the measure of the other leg is <strong>the √ 3 end root x</strong> and the measure of the hypotenuse is 2x. The perimeter of the triangle is given to be <strong>18 + 6 · the √ 3, end root</strong>, and so the equation for the perimeter can be written as <strong>2 x + x + the √ 3 end root x = 18 + 6 · the √ 3</strong>. Combining like terms and factoring out a common factor of x on the left-hand side of the equation gives <strong>(3 + the √ 3, ) · x = 18 + 6 · the √ 3</strong>. Rewriting the right-hand side of the equation by factoring out 6 gives <strong>(3 + the √ 3, ) · x = 6 · (3 + the √ 3, )</strong>. Dividing both sides of the equation by the common factor <strong>(3 + the √ 3, )</strong> gives x = 6. The longest side of the right triangle, the hypotenuse, has a length of 2x, or 2(6), which is 12.",
    hasFigure: false,
  },
  {
    id: "2be01bd9",
    type: "spr",
    questionHtml:
      "Triangle <strong>A B C</strong> is similar to triangle <strong>D E F</strong>, where angle <strong>A</strong> corresponds to angle <strong>D</strong> and angle <strong>C</strong> corresponds to angle <strong>F</strong>.  Angles <strong>C</strong> and <strong>F</strong> are right angles. If <strong>tan (A) = (50) / (7)</strong>, what is the value of  <strong>tan (E)</strong>?",
    choices: [],
    correctAnswer: ".14, 7/50",
    explanation:
      "The correct answer is <strong>seven fiftieths</strong>. It's given that triangle <strong>A B C</strong> is similar to triangle <strong>D E F</strong>, where angle <strong>A</strong> corresponds to angle <strong>D</strong> and angle <strong>C</strong> corresponds to angle <strong>F</strong>. In similar triangles, the tangents of corresponding angles are equal. Since angle <strong>A</strong> and angle <strong>D</strong> are corresponding angles, if <strong>tan (A) = (50) / (7)</strong>, then <strong>tan (D) = (50) / (7)</strong>. It's also given that angles <strong>C</strong> and <strong>F</strong> are right angles. It follows that triangle <strong>D E F</strong> is a right triangle with acute angles <strong>D</strong> and <strong>E</strong>. The tangent of one acute angle in a right triangle is the inverse of the tangent of the other acute angle in the triangle. Therefore, <strong>tan (E) = (1) / (tan (D))</strong>. Substituting <strong>(50) / (7)</strong> for <strong>tan (D)</strong> in this equation yields <strong>tan (E) = Start(1 OverOver StartFraction 50) / (7) EndEndFraction</strong>, or <strong>tan (E) = seven fiftieths</strong>. Thus, if <strong>tan (A) = (50) / (7)</strong>, the value of <strong>tan (E)</strong> is <strong>seven fiftieths</strong>. Note that 7/50 and .14 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "4c95c7d4",
    type: "mcq",
    questionHtml:
      "A graphic designer is creating a logo for a company. The logo is shown in the figure above. The logo is in the shape of a trapezoid and consists of three congruent equilateral triangles. If the perimeter of the logo is 20 centimeters, what is the combined area of the shaded regions, in square centimeters, of the logo?",
    choices: [
      { label: "A", text: "<strong>2 · the √ 3</strong>" },
      { label: "B", text: "<strong>4 · the √ 3</strong>" },
      { label: "C", text: "<strong>8 · the √ 3</strong>" },
      { label: "D", text: "<strong>16</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that the logo is in the shape of a trapezoid that consists of three congruent equilateral triangles, and that the perimeter of the trapezoid is 20 centimeters (cm). Since the perimeter of the trapezoid is the sum of the lengths of 5 of the sides of the triangles, the length of each side of an equilateral triangle is <strong>the fraction 20 over 5 = 4 centimeters</strong>. Dividing up one equilateral triangle into two right triangles yields a pair of congruent 30°-60°-90° triangles. The shorter leg of each right triangle is half the length of the side of an equilateral triangle, or 2 cm. Using the Pythagorean Theorem, <strong>a, ² + b² = c²</strong>, the height of the equilateral triangle can be found. Substituting <strong>a = 2</strong> and <strong>c = 4</strong> and solving for b yields <strong>the √, 4² − 2², end root = the √ 12, which = 2 · the √ 3 centimeters</strong> cm. The area of one equilateral triangle is <strong>one half b h</strong>, where <strong>b = 2</strong> and <strong>h = 2 · the √ 3</strong>. Therefore, the area of one equilateral triangle is <strong>one half · 4 · (2 · the √ 3, ) = 4 · the √ 3 centimeters²</strong>. The shaded area consists of two such triangles, so its area is <strong>2 · 4 · the √ 3 = 8 · the √ 3 centimeters²</strong>.Alternate approach: The area of a trapezoid can be found by evaluating the expression <strong>one half · (b sub 1 + b sub 2, ) · h</strong>, where <strong>b sub 1</strong>is the length of one base, <strong>b sub 2</strong> is the length of the other base, and h is the height of the trapezoid. Substituting <strong>b sub 1 = 8</strong>, <strong>b sub 2 = 4</strong>, and <strong>h = 2 · the √ 3</strong> yields the expression <strong>one half · (8 + 4, ) · (2 · the √ 3, )</strong>, or <strong>one half · 12 · (2 · the √ 3, )</strong>, which gives an area of <strong>12 · the √ 3 centimeters²</strong> for the trapezoid. Since two-thirds of the trapezoid is shaded, the area of the shaded region is <strong>two thirds · 12 · the √ 3 = 8 · the √ 3</strong>.<br>Choice A is incorrect. This is the height of the trapezoid. Choice B is incorrect. This is the area of one of the equilateral triangles, not two. Choice D is incorrect and may result from using a height of 4 for each triangle rather than the height of <strong>2 · the √ 3</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/4c95c7d4_img1.png",
  },
  {
    id: "55bb437a",
    type: "spr",
    questionHtml:
      "In the figure above, <strong>tan of B = the fraction 3 over 4</strong>. If <strong>B C = 15</strong> and <strong>D A = 4</strong>, what is the length of <strong>D E</strong> ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 6. Since <strong>tan of B = three fourths</strong>, <strong>triangle A, B C</strong> and <strong>triangle D B E</strong> are both similar to 3-4-5 triangles. This means that they are both similar to the right triangle with sides of lengths 3, 4, and 5. Since <strong>the length of side B C = 15</strong>, which is 3 times as long as the hypotenuse of the 3-4-5 triangle, the similarity ratio of <strong>triangle A, B C</strong> to the 3-4-5 triangle is 3:1. Therefore, the length of <strong>side A, C</strong> (the side opposite to <strong>angle B</strong>) is <strong>3 · 3 = 9</strong>, and the length of <strong>side A, B</strong> (the side adjacent to <strong>angle B</strong>) is <strong>4 · 3 = 12</strong>. It is also given that <strong>the length of side D A = 4</strong>. Since <strong>the length of side A, B = the length of side D A + the length of side D B</strong> and <strong>the length of side A, B = 12</strong>, it follows that <strong>the length of side D B = 8</strong>, which means that the similarity ratio of <strong>triangle D B E</strong> to the 3-4-5 triangle is 2:1 ( <strong>side D B</strong> is the side adjacent to <strong>angle B</strong>). Therefore, the length of <strong>side D E</strong>, which is the side opposite to <strong>angle B</strong>, is <strong>3 · 2 = 6</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/55bb437a_img1.png",
  },
  {
    id: "568d66a7",
    type: "mcq",
    questionHtml:
      "An isosceles right triangle has a perimeter of <strong>94 + 94 √(2)</strong> inches. What is the length, in inches, of one leg of this triangle?",
    choices: [
      { label: "A", text: "<strong>47</strong>" },
      { label: "B", text: "<strong>47 √(2)</strong>" },
      { label: "C", text: "<strong>94</strong>" },
      { label: "D", text: "<strong>94 √(2)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It's given that the right triangle is isosceles. In an isosceles right triangle, the two legs have equal lengths, and the length of the hypotenuse is <strong>√(2)</strong> times the length of one of the legs. Let <strong>script l</strong> represent the length, in inches, of each leg of the isosceles right triangle. It follows that the length of the hypotenuse is <strong>script l √(2)</strong> inches. The perimeter of a figure is the sum of the lengths of the sides of the figure. Therefore, the perimeter of the isosceles right triangle is <strong>script l + script l + script l √(2)</strong> inches. It's given that the perimeter of the triangle is <strong>94 + 94 √(2)</strong> inches. It follows that <strong>script l + script l + script l √(2) = 94 + 94 √(2)</strong>. Factoring the left-hand side of this equation yields <strong>(1 + 1 + √(2)) script l = 94 + 94 √(2)</strong>, or <strong>(2 + √(2)) script l = 94 + 94 √(2)</strong>. Dividing both sides of this equation by <strong>2 + √(2)</strong> yields <strong>script l = (94 + 94 √(2)) / (2 + √(2))</strong>. Rationalizing the denominator of the right-hand side of this equation by multiplying the right-hand side of the equation by <strong>(2 − √(2)) / (2 − √(2))</strong> yields <strong>script l = ((94 + 94 √(2)) (2 − √(2))) / ((2 + √(2)) (2 − √(2)))</strong>. Applying the distributive property to the numerator and to the denominator of the right-hand side of this equation yields <strong>script l = (188 − 94 √(2) + 188 √(2) − 94 √(4)) / (4 − 2 √(2) + 2 √(2) − √(4))</strong>. This is equivalent to <strong>script l = (94 √(2)) / (2)</strong>, or <strong>script l = 47 √(2)</strong>. Therefore, the length, in inches, of one leg of the isosceles right triangle is <strong>47 √(2)</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect. This is the length, in inches, of the hypotenuse.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "6933b3d9",
    type: "spr",
    questionHtml:
      "<strong>The figure presents right triangle R S T such that side R T is horizontal, vertex T is to the right of vertex R, and vertex S is above R T. Side R S is labeled 12. Side S T is labeled 5. Angle S is a right angle</strong>In triangle RST above, point W (not shown) lies on <strong>R T</strong>. What is the value of <strong>cos of angle R S W − sin of angle W S T</strong> ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 0. Note that no matter where point W is on <strong>side R T</strong>, the sum of the measures of <strong>angle R S W</strong> and <strong>angle W S T</strong> is equal to the measure of <strong>angle R S T</strong>, which is <strong>90 °</strong>. Thus, <strong>angle R S W</strong> and <strong>angle W S T</strong> are complementary angles. Since the cosine of an angle is equal to the sine of its complementary angle, <strong>the cos of angle R S W = the sin of angle W S T</strong>. Therefore, <strong>the cos of angle R S W − the sin of angle W S T = 0</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/6933b3d9_img1.png",
  },
  {
    id: "6ab30ce3",
    type: "mcq",
    questionHtml:
      "Triangle <strong>A B C</strong> is similar to triangle <strong>D E F</strong>, where <strong>A</strong> corresponds to <strong>D</strong> and <strong>C</strong> corresponds to <strong>F</strong>. Angles <strong>C</strong> and <strong>F</strong> are right angles. If <strong>tan (A) = √(3)</strong> and <strong>D F = 125</strong>, what is the length of <strong>D E</strong>?",
    choices: [
      { label: "A", text: "<strong>125 (√(3)) / (3)</strong>" },
      { label: "B", text: "<strong>125 (√(3)) / (2)</strong>" },
      { label: "C", text: "<strong>125 √(3)</strong>" },
      { label: "D", text: "<strong>250</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Corresponding angles in similar triangles have equal measures. It's given that triangle <strong>A B C</strong> is similar to triangle <strong>D E F</strong>, where <strong>A</strong> corresponds to <strong>D</strong>, so the measure of angle <strong>A</strong> is equal to the measure of angle <strong>D</strong>. Therefore, if <strong>tan (A) = √(3)</strong>, then <strong>tan (D) = √(3)</strong>. It's given that angles <strong>C</strong> and <strong>F</strong> are right angles, so triangles <strong>A B C</strong> and <strong>D E F</strong> are right triangles. The adjacent side of an acute angle in a right triangle is the side closest to the angle that is not the hypotenuse. It follows that the adjacent side of angle <strong>D</strong> is side <strong>D F</strong>. The opposite side of an acute angle in a right triangle is the side across from the acute angle. It follows that the opposite side of angle <strong>D</strong> is side <strong>E F</strong>. The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, <strong>tan (D) = (E F) / (D F)</strong>. If <strong>D F = 125</strong>, the length of side <strong>E F</strong> can be found by substituting <strong>√(3)</strong> for <strong>tan (D)</strong> and <strong>125</strong> for <strong>D F</strong> in the equation <strong>tan (D) = (E F) / (D F)</strong>, which yields <strong>√(3) = (E F) / (125)</strong>. Multiplying both sides of this equation by <strong>125</strong> yields <strong>125 √(3) = E F</strong>. Since the length of side <strong>E F</strong> is <strong>√(3)</strong> times the length of side <strong>D F</strong>, it follows that triangle <strong>D E F</strong> is a special right triangle with angle measures <strong>30 °</strong>, <strong>60 °</strong>, and <strong>90 °</strong>. Therefore, the length of the hypotenuse, <strong>D E</strong>, is <strong>2</strong> times the length of side <strong>D F</strong>, or <strong>D E = 2 (D F)</strong>. Substituting <strong>125</strong> for <strong>D F</strong> in this equation yields <strong>D E = 2 (125)</strong>, or <strong>D E = 250</strong>. Thus, if <strong>tan (A) = √(3)</strong> and <strong>D F = 125</strong>, the length of <strong>D E</strong> is <strong>250</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect. This is the length of <strong>E F</strong>, not <strong>D E</strong>.",
    hasFigure: false,
  },
  {
    id: "7c25b0dc",
    type: "spr",
    questionHtml:
      "The length of a rectangle’s diagonal is <strong>3 √(17)</strong>, and the length of the rectangle’s shorter side is <strong>3</strong>. What is the length of the rectangle’s longer side?",
    choices: [],
    correctAnswer: "12",
    explanation:
      "The correct answer is <strong>12</strong>. The diagonal of a rectangle forms a right triangle, where the shorter side and the longer side of the rectangle are the legs of the triangle and the diagonal of the rectangle is the hypotenuse of the triangle. It's given that the length of the rectangle's diagonal is <strong>3 √(17)</strong> and the length of the rectangle's shorter side is <strong>3</strong>. Thus, the length of the hypotenuse of the right triangle formed by the diagonal is <strong>3 √(17)</strong> and the length of one of the legs is <strong>3</strong>. By the Pythagorean theorem, if a right triangle has a hypotenuse with length <strong>c</strong> and legs with lengths <strong>a</strong> and <strong>b</strong>, then <strong>a² + b² = c²</strong>. Substituting <strong>3 √(17)</strong> for <strong>c</strong> and <strong>3</strong> for <strong>b</strong> in this equation yields <strong>a² + (3)² = (3 √(17))²</strong>, or <strong>a² + 9 = 153</strong>. Subtracting <strong>9</strong> from both sides of this equation yields <strong>a² = 144</strong>. Taking the square root of both sides of this equation yields <strong>a = + or − √(144)</strong>, or <strong>a = + or − 12</strong>. Since <strong>a</strong> represents a length, which must be positive, the value of <strong>a</strong> is <strong>12</strong>. Thus, the length of the rectangle's longer side is <strong>12</strong>.",
    hasFigure: false,
  },
  {
    id: "8027db3f",
    type: "spr",
    questionHtml:
      "In triangle <strong>J K L</strong>, <strong>cos (K) = (24) / (51)</strong> and angle <strong>J</strong> is a right angle. What is the value of <strong>cos (L)</strong>?",
    choices: [],
    correctAnswer: ".8823, .8824, 15/17",
    explanation:
      "The correct answer is <strong>(15) / (17)</strong>. It's given that angle <strong>J</strong> is the right angle in triangle <strong>J K L</strong>. Therefore, the acute angles of triangle <strong>J K L</strong> are angle <strong>K</strong> and angle <strong>L</strong>. The hypotenuse of a right triangle is the side opposite its right angle. Therefore, the hypotenuse of triangle <strong>J K L</strong> is side <strong>K L</strong>. The cosine of an acute angle in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. It's given that <strong>cos (K) = (24) / (51)</strong>. This can be written as <strong>cos (K) = eight seventeenths</strong>. Since the cosine of angle <strong>K</strong> is a ratio, it follows that the length of the side adjacent to angle <strong>K</strong> is <strong>8 n</strong> and the length of the hypotenuse is <strong>17 n</strong>, where <strong>n</strong> is a constant. Therefore, <strong>J K = 8 n</strong> and <strong>K L = 17 n</strong>. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. For triangle <strong>J K L</strong>, it follows that <strong>(J K)² + (J L)² = (K L)²</strong>. Substituting <strong>8 n</strong> for <strong>J K</strong> and <strong>17 n</strong> for <strong>K L</strong> yields <strong>(8 n)² + (J L)² = (17 n)²</strong>. This is equivalent to <strong>64 n² + (J L)² = 289 n²</strong>. Subtracting <strong>64 n²</strong> from each side of this equation yields <strong>(J L)² = 225 n²</strong>. Taking the square root of each side of this equation yields <strong>J L = 15 n</strong>. Since <strong>cos (L) = (J L) / (K L)</strong>, it follows that <strong>cos (L) = (15 n) / (17 n)</strong>, which can be rewritten as <strong>cos (L) = (15) / (17)</strong>. Note that 15/17, .8824, .8823, and 0.882 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "92eb236a",
    type: "mcq",
    questionHtml:
      "In a right triangle, the tangent of one of the two acute angles is <strong>the fraction with numerator the √ 3, and denominator 3</strong>. What is the tangent of the other acute angle?",
    choices: [
      {
        label: "A",
        text: "<strong>the − of the fraction with numerator the √ 3, and denominator 3</strong>",
      },
      {
        label: "B",
        text: "<strong>the − of the fraction with numerator 3, and denominator the √ 3</strong>",
      },
      {
        label: "C",
        text: "<strong>the fraction with numerator the √ 3, and denominator 3</strong>",
      },
      {
        label: "D",
        text: "<strong>the fraction with numerator 3, and denominator the √ 3</strong>",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The tangent of a nonright angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. Using that definition for tangent, in a right triangle with legs that have lengths a and b, the tangent of one acute angle is <strong>the fraction a, over b</strong> and the tangent for the other acute angle is <strong>the fraction b over a</strong>. It follows that the tangents of the acute angles in a right triangle are reciprocals of each other. Therefore, the tangent of the other acute angle in the given triangle is the reciprocal of <strong>the fraction, the √ 3, end root, over 3, end fraction</strong> or <strong>the fraction, 3 over the √ 3, end fraction</strong>.Choice A is incorrect and may result from assuming that the tangent of the other acute angle is the negative of the tangent of the angle described. Choice B is incorrect and may result from assuming that the tangent of the other acute angle is the negative of the reciprocal of the tangent of the angle described. Choice C is incorrect and may result from interpreting the tangent of the other acute angle as equal to the tangent of the angle described.",
    hasFigure: false,
  },
  {
    id: "a4bd60a3",
    type: "spr",
    questionHtml:
      "The perimeter of an equilateral triangle is <strong>624</strong> centimeters. The height of this triangle is <strong>k √(3)</strong> centimeters, where <strong>k</strong> is a constant. What is the value of <strong>k</strong>?",
    choices: [],
    correctAnswer: "104",
    explanation:
      "The correct answer is <strong>104</strong>. An equilateral triangle is a triangle in which all three sides have the same length and all three angles have a measure of <strong>60 °</strong>. The height of the triangle, <strong>k √(3)</strong>, is the length of the altitude from one vertex. The altitude divides the equilateral triangle into two congruent 30-60-90 right triangles, where the altitude is the side across from the <strong>60 °</strong> angle in each 30-60-90 right triangle. Since the altitude has a length of <strong>k √(3)</strong>, it follows from the properties of 30-60-90 right triangles that the side across from each <strong>30 °</strong> angle has a length of <strong>k</strong> and each hypotenuse has a length of <strong>2 k</strong>. In this case, the hypotenuse of each 30-60-90 right triangle is a side of the equilateral triangle; therefore, each side length of the equilateral triangle is <strong>2 k</strong>. The perimeter of a triangle is the sum of the lengths of each side. It's given that the perimeter of the equilateral triangle is <strong>624</strong>; therefore, <strong>2 k + 2 k + 2 k = 624</strong>, or <strong>6 k = 624</strong>. Dividing both sides of this equation by <strong>6</strong> yields <strong>k = 104</strong>.",
    hasFigure: false,
  },
  {
    id: "ae041e52",
    type: "mcq",
    questionHtml:
      "A square is inscribed in a circle. The radius of the circle is <strong>(20 √(2)) / (2)</strong> inches. What is the side length, in inches, of the square?",
    choices: [
      { label: "A", text: "<strong>20</strong>" },
      { label: "B", text: "<strong>(20 √(2)) / (2)</strong>" },
      { label: "C", text: "<strong>20 √(2)</strong>" },
      { label: "D", text: "<strong>40</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. When a square is inscribed in a circle, a diagonal of the square is a diameter of the circle. It's given that a square is inscribed in a circle and the length of a radius of the circle is <strong>(20 √(2)) / (2)</strong> inches. Therefore, the length of a diameter of the circle is <strong>2 ((20 √(2)) / (2))</strong> inches, or <strong>20 √(2)</strong> inches. It follows that the length of a diagonal of the square is <strong>20 √(2)</strong> inches. A diagonal of a square separates the square into two right triangles in which the legs are the sides of the square and the hypotenuse is a diagonal. Since a square has <strong>4</strong> congruent sides, each of these two right triangles has congruent legs and a hypotenuse of length <strong>20 √(2)</strong> inches. Since each of these two right triangles has congruent legs, they are both <strong>45</strong>-<strong>45</strong>-<strong>90</strong> triangles. In a <strong>45</strong>-<strong>45</strong>-<strong>90</strong> triangle, the length of the hypotenuse is <strong>√(2)</strong> times the length of a leg. Let <strong>s</strong> represent the length of a leg of one of these <strong>45</strong>-<strong>45</strong>-<strong>90</strong> triangles. It follows that <strong>20 √(2) = √(2) (s)</strong>. Dividing both sides of this equation by <strong>√(2)</strong> yields <strong>20 = s</strong>. Therefore, the length of a leg of one of these <strong>45</strong>-<strong>45</strong>-<strong>90</strong> triangles is <strong>20</strong> inches. Since the legs of these two <strong>45</strong>-<strong>45</strong>-<strong>90</strong> triangles are the sides of the square, it follows that the side length of the square is <strong>20</strong> inches.<br>Choice B is incorrect. This is the length of a radius, in inches, of the circle.<br>Choice C is incorrect. This is the length of a diameter, in inches, of the circle.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "bd87bc09",
    type: "spr",
    questionHtml:
      "Triangle <strong>A, B C</strong> above is a right triangle, and <strong>the sin of B = 5 over 13</strong>. What is the length of side <strong>B C</strong>?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 24. The sine of an acute angle in a right triangle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the triangle shown, the sine of angle B, or <strong>sin of B</strong>, is equal to the ratio of the length of side <strong>A, C</strong> to the length of side <strong>A, B</strong>. It’s given that the length of side <strong>A, B</strong> is 26 and that <strong>sin of B = 5 over 13</strong>. Therefore, <strong>the fraction 5 over 13 = the fraction A, B over 26</strong>. Multiplying both sides of this equation by 26 yields <strong>A, C = 10</strong>.By the Pythagorean Theorem, the relationship between the lengths of the sides of triangle ABC is as follows: <strong>26² = 10² + B C²</strong>, or <strong>676 = 100 + B C²</strong>. Subtracting 100 from both sides of <strong>676 = 100 + B C²</strong> yields <strong>576 = B C²</strong>. Taking the square root of both sides of <strong>576 = B C²</strong> yields <strong>24 = B C</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/bd87bc09_img1.png",
  },
  {
    id: "c6dff223",
    type: "spr",
    questionHtml:
      "Triangle <strong>A B C</strong> is similar to triangle <strong>D E F</strong>, where angle <strong>A</strong> corresponds to angle <strong>D</strong> and angles <strong>C</strong> and <strong>F</strong> are right angles. The length of <strong>A B</strong> is <strong>2.9</strong> times the length of <strong>D E</strong>. If <strong>tan A = (21) / (20)</strong>, what is the value of <strong>sin D</strong>?",
    choices: [],
    correctAnswer: ".7241, 21/29",
    explanation:
      "The correct answer is <strong>(21) / (29)</strong>. It's given that triangle <strong>A B C</strong> is similar to triangle <strong>D E F</strong>, where angle <strong>A</strong> corresponds to angle <strong>D</strong> and angles <strong>C</strong> and <strong>F</strong> are right angles. In similar triangles, the tangents of corresponding angles are equal. Therefore, if <strong>tan A = (21) / (20)</strong>, then <strong>tan D = (21) / (20)</strong>. In a right triangle, the tangent of an acute angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. Therefore, in triangle <strong>D E F</strong>, if <strong>tan D = (21) / (20)</strong>, the ratio of the length of <strong>E F</strong> to the length of <strong>D F</strong> is <strong>(21) / (20)</strong>. If the lengths of <strong>E F</strong> and <strong>D F</strong> are <strong>21</strong> and <strong>20</strong>, respectively, then the ratio of the length of <strong>E F</strong> to the length of <strong>D F</strong> is <strong>(21) / (20)</strong>. In a right triangle, the sine of an acute angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. Therefore, the value of <strong>sin D</strong> is the ratio of the length of <strong>E F</strong> to the length of <strong>D E</strong>. The length of <strong>D E</strong> can be calculated using the Pythagorean theorem, which states that if the lengths of the legs of a right triangle are <strong>a</strong> and <strong>b</strong> and the length of the hypotenuse is <strong>c</strong>, then <strong>a² + b² = c²</strong>. Therefore, if the lengths of <strong>E F</strong> and <strong>D F</strong> are <strong>21</strong> and <strong>20</strong>, respectively, then <strong>(21)² + (20)² = (D E)²</strong>, or <strong>841 = (D E)²</strong>. Taking the positive square root of both sides of this equation yields <strong>29 = D E</strong>. Therefore, if the lengths of <strong>E F</strong> and <strong>D F</strong> are <strong>21</strong> and <strong>20</strong>, respectively, then the length of <strong>D E</strong> is <strong>29</strong> and the ratio of the length of <strong>E F</strong> to the length of <strong>D E</strong> is <strong>(21) / (29)</strong>. Thus, if <strong>tan A = (21) / (20)</strong>, the value of <strong>sin D</strong> is <strong>(21) / (29)</strong>. Note that 21/29, .7241, and 0.724 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "c9931030",
    type: "mcq",
    questionHtml:
      "<strong>R S = 20</strong><br><strong>S T = 48</strong><br><strong>T R = 52</strong><br>The side lengths of right triangle <strong>R S T</strong> are given. Triangle <strong>R S T</strong> is similar to triangle <strong>U V W</strong>, where <strong>S</strong> corresponds to <strong>V</strong> and <strong>T</strong> corresponds to <strong>W</strong>. What is the value of <strong>tan W</strong>?",
    choices: [
      { label: "A", text: "<strong>five thirteenths</strong>" },
      { label: "B", text: "<strong>five twelfths</strong>" },
      { label: "C", text: "<strong>(12) / (13)</strong>" },
      { label: "D", text: "<strong>(12) / (5)</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It's given that right triangle <strong>R S T</strong> is similar to triangle <strong>U V W</strong>, where <strong>S</strong> corresponds to <strong>V</strong> and <strong>T</strong> corresponds to <strong>W</strong>. It's given that the side lengths of the right triangle <strong>R S T</strong> are <strong>R S = 20</strong>, <strong>S T = 48</strong>, and <strong>T R = 52</strong>. Corresponding angles in similar triangles are equal. It follows that the measure of angle <strong>T</strong> is equal to the measure of angle <strong>W</strong>. The hypotenuse of a right triangle is the longest side. It follows that the hypotenuse of triangle <strong>R S T</strong> is side <strong>T R</strong>. The hypotenuse of a right triangle is the side opposite the right angle. Therefore, angle <strong>S</strong> is a right angle. The adjacent side of an acute angle in a right triangle is the side closest to the angle that is not the hypotenuse. It follows that the adjacent side of angle <strong>T</strong> is side <strong>S T</strong>. The opposite side of an acute angle in a right triangle is the side across from the acute angle. It follows that the opposite side of angle <strong>T</strong> is side <strong>R S</strong>. The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, <strong>tan T = (R S) / (S T)</strong>. Substituting <strong>20</strong> for <strong>R S</strong> and <strong>48</strong> for <strong>S T</strong> in this equation yields <strong>tan T = (20) / (48)</strong>, or <strong>tan T = five twelfths</strong>. The tangents of two acute angles with equal measures are equal. Since the measure of angle <strong>T</strong> is equal to the measure of angle <strong>W</strong>, it follows that <strong>tan T = tan W</strong>. Substituting <strong>five twelfths</strong> for <strong>tan T</strong> in this equation yields <strong>five twelfths = tan W</strong>. Therefore, the value of <strong>tan W</strong> is <strong>five twelfths</strong>.<br>Choice A is incorrect. This is the value of <strong>sin W</strong>.<br>Choice C is incorrect. This is the value of <strong>cos W</strong>.<br>Choice D is incorrect. This is the value of <strong>(1) / (tan W)</strong>.",
    hasFigure: false,
  },
  {
    id: "dba6a25a",
    type: "spr",
    questionHtml:
      "In the figure above, <strong>B D</strong> is parallel to <strong>A, E</strong>. What is the length of <strong>C E</strong> ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 30. In the figure given, since <strong>side B D</strong> is parallel to <strong>side A, E</strong> and both segments are intersected by <strong>side C E</strong>, then angle BDC and angle AEC are corresponding angles and therefore congruent. Angle BCD and angle ACE are also congruent because they are the same angle. Triangle BCD and triangle ACE are similar because if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Since triangle BCD and triangle ACE are similar, their corresponding sides are proportional. So in triangle BCD and triangle ACE, <strong>side B D</strong> corresponds to <strong>side A, E</strong> and <strong>side C D</strong> corresponds to <strong>side C E</strong>. Therefore, <strong>the length of side B D over the length of side C D = the length of side A, E over the length of side C E</strong>. Since triangle BCD is a right triangle, the Pythagorean theorem can be used to give the value of CD: <strong>6² + 8² = the length of side C D²</strong>. Taking the square root of each side gives <strong>the length of side C D = 10</strong>. Substituting the values in the proportion <strong>the length of side B D over the length of side C D = the length of side A, E over the length of side C E</strong> yields <strong>6 over 10 = 18 over the length of C E</strong>. Multiplying each side by CE, and then multiplying by <strong>10 over 6</strong> yields <strong>the length of side C E equal to 30</strong>. Therefore, the length of <strong>side C E</strong> is 30.",
    hasFigure: true,
    figureUrl: "/practice-images/dba6a25a_img1.png",
  },
  {
    id: "f811d345",
    type: "spr",
    questionHtml:
      "A right triangle has legs with lengths of <strong>24</strong> centimeters and <strong>21</strong> centimeters. If the length of this triangle's hypotenuse, in centimeters, can be written in the form <strong>3 √(d)</strong>, where <strong>d</strong> is an integer, what is the value of <strong>d</strong>?",
    choices: [],
    correctAnswer: "113",
    explanation:
      "The correct answer is <strong>113</strong>. It's given that the legs of a right triangle have lengths <strong>24</strong> centimeters and <strong>21</strong> centimeters. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs. It follows that if <strong>h</strong> represents the length, in centimeters, of the hypotenuse of the right triangle, <strong>h² = 24² + 21²</strong>. This equation is equivalent to <strong>h² = 1, 017</strong>. Taking the square root of each side of this equation yields <strong>h = √(1, 017)</strong>. This equation can be rewritten as <strong>h = √(9 dot 113)</strong>, or <strong>h = √(9) dot √(113)</strong>. This equation is equivalent to <strong>h = 3 √(113)</strong>. It's given that the length of the triangle's hypotenuse, in centimeters, can be written in the form <strong>3 √(d)</strong>. It follows that the value of <strong>d</strong> is <strong>113</strong>.",
    hasFigure: false,
  },
  {
    id: "ffe862a3",
    type: "mcq",
    questionHtml:
      "An isosceles right triangle has a hypotenuse of length <strong>58</strong> inches. What is the perimeter, in inches, of this triangle?",
    choices: [
      { label: "A", text: "<strong>29 √(2)</strong>" },
      { label: "B", text: "<strong>58 √(2)</strong>" },
      { label: "C", text: "<strong>58 + 58 √(2)</strong>" },
      { label: "D", text: "<strong>58 + 116 √(2)</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Since the triangle is an isosceles right triangle, the two sides that form the right angle must be the same length. Let <strong>x</strong> be the length, in inches, of each of those sides. The Pythagorean theorem states that in a right triangle, <strong>a² + b² = c²</strong>, where <strong>c</strong> is the length of the hypotenuse and <strong>a</strong> and <strong>b</strong> are the lengths of the other two sides. Substituting <strong>x</strong> for <strong>a</strong>, <strong>x</strong> for <strong>b</strong>, and <strong>58</strong> for <strong>c</strong> in this equation yields <strong>x² + x² = 58²</strong>, or <strong>2 x² = 58²</strong>. Dividing each side of this equation by <strong>2</strong> yields <strong>x² = (58²) / (2)</strong>, or <strong>x² = (2 dot 58²) / (4)</strong>. Taking the square root of each side of this equation yields two solutions: <strong>x = (58 √(2)) / (2)</strong> and <strong>x = − (58 √(2)) / (2)</strong>. The value of <strong>x</strong> must be positive because it represents a side length. Therefore, <strong>x = (58 √(2)) / (2)</strong>, or <strong>x = 29 √(2)</strong>. The perimeter, in inches, of the triangle is <strong>58 + x + x</strong>, or <strong>58 + 2 x</strong>. Substituting <strong>29 √(2)</strong> for <strong>x</strong> in this expression gives a perimeter, in inches, of <strong>58 + 2 (29 √(2))</strong>, or <strong>58 + 58 √(2)</strong>.<br>Choice A is incorrect. This is the length, in inches, of each of the congruent sides of the triangle, not the perimeter, in inches, of the triangle.<br>Choice B is incorrect. This is the sum of the lengths, in inches, of the congruent sides of the triangle, not the perimeter, in inches, of the triangle.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
];
