import { type PracticeQuestion } from "../../types/lesson"; export const CIRCLES_EASY: PracticeQuestion[] = [ { id: "23c5fcce", type: "mcq", questionHtml: "The circle above with center O has a circumference of 36. What is the length of minor arc A, C?", choices: [ { label: "A", text: "9" }, { label: "B", text: "12" }, { label: "C", text: "18" }, { label: "D", text: "36" }, ], correctAnswer: "A", explanation: "Choice A is correct. A circle has 360 degrees of arc. In the circle shown, O is the center of the circle and angle A, O C is a central angle of the circle. From the figure, the two diameters that meet to form angle A, O C are perpendicular, so the measure of angle A, O C is 90 °. Therefore, the length of minor arc A, C is the fraction 90 over 360 of the circumference of the circle. Since the circumference of the circle is 36, the length of minor arc A, C is the fraction 90 over 360, end fraction · 36 = 9.Choices B, C, and D are incorrect. The perpendicular diameters divide the circumference of the circle into four equal arcs; therefore, minor arc A, C is one fourth of the circumference. However, the lengths in choices B and C are, respectively, one third and one half the circumference of the circle, and the length in choice D is the length of the entire circumference. None of these lengths is one fourth the circumference.", hasFigure: true, figureUrl: "/practice-images/23c5fcce_img1.png", }, ]; export const CIRCLES_MEDIUM: PracticeQuestion[] = [ { id: "0815a5af", type: "mcq", questionHtml: "The center of the circle is point upper O.
Points upper S, upper R, upper Q, and upper P are on the circle.
Line segment upper P upper R is a diameter of the circle.
Line segment upper Q upper S is a diameter of the circle.
Diameters upper P upper R and upper Q upper S intersect at point upper O.
A note indicates the figure is not drawn to scale.

The circle shown has center O, circumference 144 π, and diameters P R and Q S. The length of arc P S is twice the length of arc P Q. What is the length of arc Q R?", choices: [ { label: "A", text: "24 π" }, { label: "B", text: "48 π" }, { label: "C", text: "72 π" }, { label: "D", text: "96 π" }, ], correctAnswer: "B", explanation: "Choice B is correct. Since P R and Q S are diameters of the circle shown, O S, O R, O P, and O Q are radii of the circle and are therefore congruent. Since angle S O P and angle R O Q are vertical angles, they are congruent. Therefore, arc P S and arc Q R are formed by congruent radii and have the same angle measure, so they are congruent arcs. Similarly, angle S O R and angle P O Q are vertical angles, so they are congruent. Therefore, arc S R and arc P Q are formed by congruent radii and have the same angle measure, so they are congruent arcs. Let x represent the length of arc S R. Since arc S R and arc P Q are congruent arcs, the length of arc P Q can also be represented by x. It’s given that the length of arc P S is twice the length of arc P Q. Therefore, the length of arc P S can be represented by the expression 2 x. Since arc P S and arc Q R are congruent arcs, the length of arc Q R can also be represented by 2 x. This gives the expression x + x + 2 x + 2 x. Since it's given that the circumference is 144 π, the expression x + x + 2 x + 2 x is equal to 144 π. Thus x + x + 2 x + 2 x = 144 π, or 6 x = 144 π. Dividing both sides of this equation by 6 yields x = 24 π. Therefore, the length of arc Q R is 2 (24 π), or 48 π.
Choice A is incorrect. This is the length of arc P Q, not arc Q R.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.", hasFigure: true, figureUrl: "/practice-images/0815a5af_svg1.svg", }, { id: "74d8b897", type: "spr", questionHtml: "An angle has a measure of (9 π) / (20) radians. What is the measure of the angle in degrees?", choices: [], correctAnswer: "81", explanation: "The correct answer is 81. The measure of an angle, in degrees, can be found by multiplying its measure, in radians, by (180 °) / (π radians). Multiplying the given angle measure, (9 π) / (20) radians, by (180 °) / (π radians) yields ((9 π) / (20) radians) ((180 °) / (π radians)), which is equivalent to 81 degrees.", hasFigure: false, }, { id: "82c8325f", type: "mcq", questionHtml: "A circle in the xy-plane has its center at (−4, 5) and the point (−8, 8) lies on the circle. Which equation represents this circle?", choices: [ { label: "A", text: "(x − 4)² + (y + 5)² = 5" }, { label: "B", text: "(x + 4)² + (y − 5)² = 5" }, { label: "C", text: "(x − 4)² + (y + 5)² = 25" }, { label: "D", text: "(x + 4)² + (y − 5)² = 25" }, ], correctAnswer: "D", explanation: "Choice D is correct. A circle in the xy-plane can be represented by an equation of the form (x − h)² + (y − k)² = r², where (h, k) is the center of the circle and r is the length of a radius of the circle. It's given that the circle has its center at (−4, 5). Therefore, h = −4 and k = 5. Substituting −4 for h and 5 for k in the equation (x − h)² + (y − k)² = r² yields (x − (−4))² + (y − 5)² = r², or (x + 4)² + (y − 5)² = r². It's also given that the point (−8, 8) lies on the circle. Substituting −8 for x and 8 for y in the equation (x + 4)² + (y − 5)² = r² yields (−8 + 4)² + (8 − 5)² = r², or (−4)² + (3)² = r², which is equivalent to 16 + 9 = r², or 25 = r². Substituting 25 for r² in the equation (x + 4)² + (y − 5)² = r² yields (x + 4)² + (y − 5)² = 25. Thus, the equation (x + 4)² + (y − 5)² = 25 represents the circle.
Choice A is incorrect. The circle represented by this equation has its center at (4 −5), not (−4, 5), and the point (−8, 8) doesn't lie on the circle.
Choice B is incorrect. The point (−8, 8) doesn't lie on the circle represented by this equation.
Choice C is incorrect. The circle represented by this equation has its center at (4 −5), not (−4, 5), and the point (−8, 8) doesn't lie on the circle.", hasFigure: false, }, { id: "856372ca", type: "mcq", questionHtml: "In the xy-plane, a circle with radius 5 has center with coordinates − 8, 6. Which of the following is an equation of the circle?", choices: [ { label: "A", text: "(x − 8, ), ² + (y + 6, ), ² = 25" }, { label: "B", text: "(x + 8, ), ² + (y − 6, ), ² = 25" }, { label: "C", text: "(x − 8, ), ² + (y + 6, ), ² = 5" }, { label: "D", text: "(x + 8, ), ² + (y − 6, ), ² = 5" }, ], correctAnswer: "B", explanation: "Choice B is correct. An equation of a circle is (x − h, ), ² + (y − k, ), ² = r², where the center of the circle is h, k and the radius is r. It’s given that the center of this circle is −8, 6 and the radius is 5. Substituting these values into the equation gives (x − −8, ), ² + (y − 6, ), ² = 5², or (x + 8, ), ² + (y − 6, ), ² = 25.Choice A is incorrect. This is an equation of a circle that has center 8 −6. Choice C is incorrect. This is an equation of a circle that has center 8 −6 and radius the √ 5. Choice D is incorrect. This is an equation of a circle that has radius  the √ 5.", hasFigure: false, }, { id: "8e7689e0", type: "spr", questionHtml: "The number of radians in a 720-degree angle can be written as a · π, where a is a constant. What is the value of a ?", choices: [], correctAnswer: "", explanation: "The correct answer is 4. There are π radians in a 180 ° angle. An angle measure of 720 ° is 4 times greater than an angle measure of 180 °. Therefore, the number of radians in a 720 ° angle is 4 π.", hasFigure: false, }, { id: "95ba2d09", type: "mcq", questionHtml: "In the xy-plane above, points P, Q, R, and T lie on the circle with center O. The degree measures of angles P O Q and R O T are each 30°. What is the radian measure of angle Q O R ?", choices: [ { label: "A", text: "five sixths, π" }, { label: "B", text: "three fourths, π" }, { label: "C", text: "two thirds, π" }, { label: "D", text: "one third, π" }, ], correctAnswer: "C", explanation: "Choice C is correct. Because points T, O, and P all lie on the x-axis, they form a line. Since the angles on a line add up to 180 °, and it’s given that angles POQ and ROT each measure 30 °, it follows that the measure of angle QOR is 180 ° − 30 ° − 30 ° = 120 °. Since the arc of a complete circle is 360 ° or 2 π radians, a proportion can be set up to convert the measure of angle QOR from degrees to radians: the fraction 360 ° over 2 π radians = the fraction 120 ° over x radians, where x is the radian measure of angle QOR. Multiplying each side of the proportion by 2 π x gives 360 x = 240 π. Solving for x gives the fraction 240 over 360 · π, or two thirds π.Choice A is incorrect and may result from subtracting only angle POQ from 180 °to get a value of 150 °and then finding the radian measure equivalent to that value. Choice B is incorrect and may result from a calculation error. Choice D is incorrect and may result from calculating the sum of the angle measures, in radians, of angles POQ and ROT.", hasFigure: true, figureUrl: "/practice-images/95ba2d09_img1.png", }, { id: "a0cacec1", type: "spr", questionHtml: "An angle has a measure of (16 π) / (15) radians. What is the measure of the angle, in degrees?", choices: [], correctAnswer: "192", explanation: "The correct answer is 192. The measure of an angle, in degrees, can be found by multiplying its measure, in radians, by (180 °) / (π radians). Multiplying the given angle measure, (16 π) / (15) radians, by (180 °) / (π radians) yields ((16 π) / (15) radians) ((180 °) / (π r a d i a n s)), which simplifies to 192 degrees.", hasFigure: false, }, { id: "f1c1e971", type: "mcq", questionHtml: "The measure of angle R is (2 π) / (3) radians. The measure of angle T is (5 π) / (12) radians greater than the measure of angle R. What is the measure of angle T, in degrees?", choices: [ { label: "A", text: "75" }, { label: "B", text: "120" }, { label: "C", text: "195" }, { label: "D", text: "390" }, ], correctAnswer: "C", explanation: "Choice C is correct. It’s given that the measure of angle R is (2 π) / (3) radians, and the measure of angle T is (5 π) / (12) radians greater than the measure of angle R. Therefore, the measure of angle T is equal to (2 π) / (3) + (5 π) / (12) radians. Multiplying (2 π) / (3) by four fourths to get a common denominator with (5 π) / (12) yields (8 π) / (12). Therefore, (2 π) / (3) + (5 π) / (12) is equivalent to (8 π) / (12) + (5 π) / (12), or (13 π) / (12). Therefore, the measure of angle T is (13 π) / (12) radians. The measure of angle T, in degrees, can be found by multiplying its measure, in radians, by (180) / (π). This yields (13 π) / (12) · (180) / (π), which is equivalent to 195 degrees. Therefore, the measure of angle T is 195 degrees.
Choice A is incorrect. This is the number of degrees that the measure of angle T is greater than the measure of angle R.
Choice B is incorrect. This is the measure of angle R, in degrees.
Choice D is incorrect and may result from conceptual or calculation errors.", hasFigure: false, }, ]; export const CIRCLES_HARD: PracticeQuestion[] = [ { id: "2266984b", type: "mcq", questionHtml: "The equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle?", choices: [ { label: "A", text: "−20 −16" }, { label: "B", text: "−10 −8" }, { label: "C", text: "10, 8" }, { label: "D", text: "20, 16" }, ], correctAnswer: "B", explanation: "Choice B is correct. The standard equation of a circle in the xy-plane is of the form (x − h, ), ² + (y − k, ), ² = r², where the ordered pair h, k are the coordinates of the center of the circle and r is the radius. The given equation can be rewritten in standard form by completing the squares. So the sum of the first two terms, x² + 20 x, needs a 100 to complete the square, and the sum of the second two terms, y² + 16 y, needs a 64 to complete the square. Adding 100 and 64 to both sides of the given equation yields (x² + 20 x + 100, ) + (y² + 16 y + 64, ) = −20 + 100 + 64, which is equivalent to (x + 10, ), ² + (y + 8, ), ² = 144. Therefore, the coordinates of the center of the circle are −10 −8.Choices A, C, and D are incorrect and may result from computational errors made when attempting to complete the squares or when identifying the coordinates of the center.", hasFigure: false, }, { id: "249d3f80", type: "spr", questionHtml: "Point O is the center of a circle. The measure of arc R S on this circle is 100 °. What is the measure, in degrees, of its associated angle R O S?", choices: [], correctAnswer: "100", explanation: "The correct answer is 100. It's given that point O is the center of a circle and the measure of arc R S on the circle is 100 °. It follows that points R and S lie on the circle. Therefore, ModifyingAbove O R With bar and ModifyingAbove O S With bar are radii of the circle. A central angle is an angle formed by two radii of a circle, with its vertex at the center of the circle. Therefore, angle R O S is a central angle. Because the degree measure of an arc is equal to the measure of its associated central angle, it follows that the measure, in degrees, of angle R O S is 100.", hasFigure: false, }, { id: "24cec8d1", type: "spr", questionHtml: "A circle has center O, and points R and S lie on the circle. In triangle O R S, the measure of angle R O S is 88 °. What is the measure of angle R S O, in degrees? (Disregard the degree symbol when entering your answer.)", choices: [], correctAnswer: "46", explanation: "The correct answer is 46. It's given that O is the center of a circle and that points R and S lie on the circle. Therefore, ModifyingAbove O R With bar and ModifyingAbove O S With bar are radii of the circle. It follows that O R = O S. If two sides of a triangle are congruent, then the angles opposite them are congruent. It follows that the angles angle R S O and angle O R S, which are across from the sides of equal length, are congruent. Let x ° represent the measure of angle R S O. It follows that the measure of angle O R S is also x °. It's given that the measure of angle R O S is 88 °. Because the sum of the measures of the interior angles of a triangle is 180 °, the equation x ° + x ° + 88 ° = 180 °, or 2 x + 88 = 180, can be used to find the measure of angle R S O. Subtracting 88 from both sides of this equation yields 2 x = 92. Dividing both sides of this equation by 2 yields x = 46. Therefore, the measure of angle R S O, in degrees, is 46.", hasFigure: false, }, { id: "3e577e4a", type: "mcq", questionHtml: "A circle in the xy-plane has its center at (−4 −6). Line k is tangent to this circle at the point (−7 −7). What is the slope of line k?", choices: [ { label: "A", text: "−3" }, { label: "B", text: "−one third" }, { label: "C", text: "one third" }, { label: "D", text: "3" }, ], correctAnswer: "A", explanation: "Choice A is correct. A line that's tangent to a circle is perpendicular to the radius of the circle at the point of tangency. It's given that the circle has its center at (−4 −6) and line k is tangent to the circle at the point (−7 −7). The slope of a radius defined by the points (q, r) and (s, t) can be calculated as (t − r) / (s − q). The points (−7 −7) and (−4 −6) define the radius of the circle at the point of tangency. Therefore, the slope of this radius can be calculated as ((−6) − (−7)) / ((−4) − (−7)), or one third. If a line and a radius are perpendicular, the slope of the line must be the negative reciprocal of the slope of the radius. The negative reciprocal of one third is −3. Thus, the slope of line k is −3.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect. This is the slope of the radius of the circle at the point of tangency, not the slope of line k.
Choice D is incorrect and may result from conceptual or calculation errors.", hasFigure: false, }, { id: "69b0d79d", type: "mcq", questionHtml: "Point O is the center of the circle above, and the measure of angle O A, B is 30 °. If the length of O C is 18, what is the length of arc A, B?", choices: [ { label: "A", text: "9 π" }, { label: "B", text: "12 π" }, { label: "C", text: "15 π" }, { label: "D", text: "18 π" }, ], correctAnswer: "B", explanation: "Choice B is correct. Because segments OA and OB are radii of the circle centered at point O, these segments have equal lengths. Therefore, triangle AOB is an isosceles triangle, where angles OAB and OBA are congruent base angles of the triangle. It’s given that angle OAB measures 30 °. Therefore, angle OBA also measures 30 °. Let x ° represent the measure of angle AOB. Since the sum of the measures of the three angles of any triangle is 180 °, it follows that 30 ° + 30 ° + x ° = 180 °, or 60 ° + x ° = 180 °. Subtracting 60 ° from both sides of this equation yields x ° = 120 °, or the fraction 2 π over 3 radians. Therefore, the measure of angle AOB, and thus the measure of arc A, B, is the fraction 2 π over 3 radians. Since the O C is a radius of the given circle and its length is 18, the length of the radius of the circle is 18. Therefore, the length of arc A, B can be calculated as the fraction 2 π over 3, end fraction · 18, or 12 π.Choices A, C, and D are incorrect and may result from conceptual or computational errors.", hasFigure: true, figureUrl: "/practice-images/69b0d79d_img1.png", }, { id: "76c73dbf", type: "spr", questionHtml: "The graph of x² + x + y² + y = (199) / (2) in the xy-plane is a circle. What is the length of the circle’s radius?", choices: [], correctAnswer: "10", explanation: "The correct answer is 10. It's given that the graph of x² + x + y² + y = (199) / (2) in the xy-plane is a circle. The equation of a circle in the xy-plane can be written in the form (x − h)² + (y − k)² = r², where the coordinates of the center of the circle are (h, k) and the length of the radius of the circle is r. The term (x − h)² in this equation can be obtained by adding the square of half the coefficient of x to both sides of the given equation to complete the square. The coefficient of x is 1. Half the coefficient of x is one half. The square of half the coefficient of x is one fourth. Adding one fourth to each side of (x² + x) + (y² + y) = (199) / (2) yields (x² + x + one fourth) + (y² + y) = (199) / (2) + one fourth, or (x + one half)² + (y² + y) = (199) / (2) + one fourth. Similarly, the term (y − k)² can be obtained by adding the square of half the coefficient of y to both sides of this equation, which yields (x + one half)² + (y² + y + one fourth) = (199) / (2) + one fourth + one fourth, or (x + one half)² + (y + one half)² = (199) / (2) + one fourth + one fourth. This equation is equivalent to (x + one half)² + (y + one half)² = 100, or (x + one half)² + (y + one half)² = 10². Therefore, the length of the circle's radius is 10.", hasFigure: false, }, { id: "89661424", type: "spr", questionHtml: "A circle in the xy-plane has its center at (−5, 2) and has a radius of 9. An equation of this circle is x² + y² + a x + b y + c = 0, where a, b, and c are constants. What is the value of c?", choices: [], correctAnswer: "-52", explanation: "The correct answer is −52. The equation of a circle in the xy-plane with its center at (h, k) and a radius of r can be written in the form (x − h)² + (y − k)² = r². It's given that a circle in the xy-plane has its center at (−5, 2) and has a radius of 9. Substituting −5 for h, 2 for k, and 9 for r in the equation (x − h)² + (y − k)² = r² yields (x − (−5))² + (y − 2)² = 9², or (x + 5)² + (y − 2)² = 81. It's also given that an equation of this circle is x² + y² + a x + b y + c = 0, where a, b, and c are constants. Therefore, (x + 5)² + (y − 2)² = 81 can be rewritten in the form x² + y² + a x + b y + c = 0. The equation (x + 5)² + (y − 2)² = 81, or (x + 5) (x + 5) + (y − 2) (y − 2) = 81, can be rewritten as x² + 5 x + 5 x + 25 + y² − 2 y − 2 y + 4 = 81. Combining like terms on the left-hand side of this equation yields x² + y² + 10 x − 4 y + 29 = 81. Subtracting 81 from both sides of this equation yields x² + y² + 10 x − 4 y − 52 = 0, which is equivalent to x² + y² + 10 x + (−4) y + (−52) = 0. This equation is in the form x² + y² + a x + b y + c = 0. Therefore, the value of c is −52.", hasFigure: false, }, { id: "981275d2", type: "mcq", questionHtml: "In the xy-plane, the graph of the equation above is a circle. Point P is on the circle and has coordinates 10 −5. If P Q is a diameter of the circle, what are the coordinates of point Q ?", choices: [ { label: "A", text: "2 −5" }, { label: "B", text: "6 −1" }, { label: "C", text: "6 −5" }, { label: "D", text: "6 −9" }, ], correctAnswer: "A", explanation: "Choice A is correct. The standard form for the equation of a circle is (x − h, ), ² + (y − k, ), ² = r², where the ordered pair h, k are the coordinates of the center and r is the length of the radius. According to the given equation, the center of the circle is 6 −5. Let x sub 1, y sub 1 represent the coordinates of point Q. Since point P 10 −5 and point Q x sub 1, y sub 1 are the endpoints of a diameter of the circle, the center with coordinates 6 −5 lies on the diameter, halfway between P and Q. Therefore, the following relationships hold: the fraction with numerator x sub 1 + 10, and denominator 2 = 6 and the fraction with numerator y sub 1 + −5, and denominator 2 = −5. Solving the equations for x sub 1 and y sub 1, respectively, yields x sub 1 = 2 and y sub 1 = −5. Therefore, the coordinates of point Q are 2 −5.Alternate approach: Since point P 10 −5 on the circle and the center of the circle 6 −5 have the same y-coordinate, it follows that the radius of the circle is 10 − 6 = 4. In addition, the opposite end of the diameter P Q must have the same y-coordinate as P and be 4 units away from the center. Hence, the coordinates of point Q must be 2 −5.
Choices B and D are incorrect because the points given in these choices lie on a diameter that is perpendicular to the diameter P Q. If either of these points were point Q, then P Q would not be the diameter of the circle. Choice C is incorrect because 6 −5 is the center of the circle and does not lie on the circle.", hasFigure: false, }, { id: "9acd101f", type: "mcq", questionHtml: "The equation x² + (y − 1)² = 49 represents circle A. Circle B is obtained by shifting circle A down 2 units in the xy-plane. Which of the following equations represents circle B?", choices: [ { label: "A", text: "(x − 2)² + (y − 1)² = 49" }, { label: "B", text: "x² + (y − 3)² = 49" }, { label: "C", text: "(x + 2)² + (y − 1)² = 49" }, { label: "D", text: "x² + (y + 1)² = 49" }, ], correctAnswer: "D", explanation: "Choice D is correct. The graph in the xy-plane of an equation of the form (x − h)² + (y − k)² = r² is a circle with center (h, k) and a radius of length r. It's given that circle A is represented by x² + (y − 1)² = 49, which can be rewritten as x² + (y − 1)² = 7². Therefore, circle A has center (0, 1) and a radius of length 7. Shifting circle A down two units is a rigid vertical translation of circle A that does not change its size or shape. Since circle B is obtained by shifting circle A down two units, it follows that circle B has the same radius as circle A, and for each point (x, y) on circle A, the point (x, y − 2) lies on circle B. Moreover, if (h, k) is the center of circle A, then (h, k − 2) is the center of circle B. Therefore, circle B has a radius of 7 and the center of circle B is (0, 1 − 2), or (0 −1). Thus, circle B can be represented by the equation x² + (y + 1)² = 7², or x² + (y + 1)² = 49.
Choice A is incorrect. This is the equation of a circle obtained by shifting circle A right 2 units.
Choice B is incorrect. This is the equation of a circle obtained by shifting circle A up 2 units.
Choice C is incorrect. This is the equation of a circle obtained by shifting circle A left 2 units.", hasFigure: false, }, { id: "9d159400", type: "mcq", questionHtml: "Which of the following equations represents a circle in the xy-plane that intersects the y-axis at exactly one point?", choices: [ { label: "A", text: "(x − 8)² + (y − 8)² = 16" }, { label: "B", text: "(x − 8)² + (y − 4)² = 16" }, { label: "C", text: "(x − 4)² + (y − 9)² = 16" }, { label: "D", text: "x² + (y − 9)² = 16" }, ], correctAnswer: "C", explanation: "Choice C is correct. The graph of the equation (x − h)² + (y − k)² = r² in the xy-plane is a circle with center (h, k) and a radius of length r. The radius of a circle is the distance from the center of the circle to any point on the circle. If a circle in the xy-plane intersects the y-axis at exactly one point, then the perpendicular distance from the center of the circle to this point on the y-axis must be equal to the length of the circle's radius. It follows that the x-coordinate of the circle's center must be equivalent to the length of the circle's radius. In other words, if the graph of (x − h)² + (y − k)² = r² is a circle that intersects the y-axis at exactly one point, then r = |h| must be true. The equation in choice C is (x − 4)² + (y − 9)² = 16, or (x − 4)² + (y − 9)² = 4². This equation is in the form (x − h)² + (y − k)² = r², where h = 4, k = 9, and r = 4, and represents a circle in the xy-plane with center (4, 9) and radius of length 4. Substituting 4 for r and 4 for h in the equation r = |h| yields 4 = |4|, or 4 = 4, which is true. Therefore, the equation in choice C represents a circle in the xy-plane that intersects the y-axis at exactly one point. 
Choice A is incorrect. This is the equation of a circle that does not intersect the y-axis at any point.
Choice B is incorrect. This is an equation of a circle that intersects the x-axis, not the y-axis, at exactly one point.
Choice D is incorrect. This is the equation of a circle with the center located on the y-axis and thus intersects the y-axis at exactly two points, not exactly one point.", hasFigure: false, }, { id: "9e44284b", type: "mcq", questionHtml: "In the xy-plane, the graph of 2 x² − 6 x + 2 y² + 2 y = 45 is a circle. What is the radius of the circle?", choices: [ { label: "A", text: "5" }, { label: "B", text: "6.5" }, { label: "C", text: "√ 40" }, { label: "D", text: "√ 50" }, ], correctAnswer: "A", explanation: "Choice A is correct. One way to find the radius of the circle is to rewrite the given equation in standard form, (x − h, ), ² + (y − k, ), ² = r², where the ordered pair h, k is the center of the circle and the radius of the circle is r. To do this, divide the original equation, 2 x² − 6 x + 2 y² + 2 y = 45, by 2 to make the leading coefficients of x² and y² each equal to 1: as follows: x² − 3 x + y² + y = 22 . 5. Then complete the square to put the equation in standard form. To do so, first rewrite x² − 3 x + y² + y = 22 . 5 as (x² − 3 x + 2 . 2 5, ) − 2 . 2 5 + (y² + y + 0 . 2 5, ) − 0 . 2 5 = 22 . 5. Second, add 2.25 and 0.25 to both sides of the equation: (x² − 3 x + 2 . 2 5, ) + (y² + y + 0 . 2 5, ) = 25. Since x² − 3 x + 2 . 2 5 = (x − 1 . 5, ), ², y² + y + 0 . 2 5 = (y + 0 . 5, ), ², and 25 = 5², it follows that (x − 1 . 5, ), ² + (y + 0 . 5, ), ² = 5². Therefore, the radius of the circle is 5.Choices B, C, and D are incorrect and may be the result of errors in manipulating the equation or of a misconception about the standard form of the equation of a circle in the xy-plane.", hasFigure: false, }, { id: "ab176ad6", type: "spr", questionHtml: "The equation (x + 6, ), ² + (y + 3, ), ² = 121 defines a circle in the xy‑plane. What is the radius of the circle?", choices: [], correctAnswer: "", explanation: "The correct answer is 11. A circle with equation (x − a, ), ² + (y − b, ), ² = r², where a, b, and r are constants, has center with coordinates a, , b and radius r. Therefore, the radius of the given circle is the √ 121, or 11.", hasFigure: false, }, { id: "acd30391", type: "mcq", questionHtml: "A circle in the xy-plane has equation (x + 3, ), ² + (y − 1, ), ² = 25. Which of the following points does NOT lie in the interior of the circle?", choices: [ { label: "A", text: "−7, 3" }, { label: "B", text: "−3, 1" }, { label: "C", text: "zero, zero" }, { label: "D", text: "3, 2" }, ], correctAnswer: "D", explanation: "Choice D is correct. The circle with equation (x + 3, ), ² + (y − 1, ), ² = 25 has center with coordinates − 3, 1 and radius 5. For a point to be inside of the circle, the distance from that point to the center must be less than the radius, 5. The distance between 3, 2 and −3, 1 is the √, (−3 − 3, ), ² + (1 − 2, ), ², end root = the √, (−6, ), ² + (−1, ), ², end root, which = the √ 37, which is greater than 5. Therefore, 3, 2 does NOT lie in the interior of the circle.Choice A is incorrect. The distance between −7, 3 and −3, 1 is the √, (−7 + 3, ), ² + (3 − 1, ), ², end root = the √, (−4, ), ² + (2, ), ², end root, which = the √ 20, which is less than 5, and therefore −7, 3 lies in the interior of the circle. Choice B is incorrect because it is the center of the circle. Choice C is incorrect because the distance between 0, 0 and −3, 1 is the √, (0 + 3, ), ² + (0 − 1, ), ², end root = the √, (3, ), ² + (1, ), ², end root, which = the √ 8, which is less than 5, and therefore 0, 0 in the interior of the circle.", hasFigure: false, }, { id: "b0a72bdc", type: "mcq", questionHtml: "What is the diameter of the circle in the xy-plane with equation (x − 5)² + (y − 3)² = 16?", choices: [ { label: "A", text: "4" }, { label: "B", text: "8" }, { label: "C", text: "16" }, { label: "D", text: "32" }, ], correctAnswer: "B", explanation: "Choice B is correct. The standard form of an equation of a circle in the xy-plane is (x − h)² + (y − k)² = r², where the coordinates of the center of the circle are (h, k) and the length of the radius of the circle is r. For the circle in the xy-plane with equation (x − 5)² + (y − 3)² = 16, it follows that r² = 16. Taking the square root of both sides of this equation yields r = 4 or r = −4. Because r represents the length of the radius of the circle and this length must be positive, r = 4. Therefore, the radius of the circle is 4. The diameter of a circle is twice the length of the radius of the circle. Thus, 2 (4) yields 8. Therefore, the diameter of the circle is 8.
Choice A is incorrect. This is the radius of the circle. 
Choice C is incorrect. This is the square of the radius of the circle. 
Choice D is incorrect and may result from conceptual or calculation errors.", hasFigure: false, }, { id: "b8a225ff", type: "spr", questionHtml: "Circle A in the xy-plane has the equation (x + 5)² + (y − 5)² = 4. Circle B has the same center as circle A. The radius of circle B is two times the radius of circle A. The equation defining circle B in the xy-plane is (x + 5)² + (y − 5)² = k, where k is a constant. What is the value of k?", choices: [], correctAnswer: "16", explanation: "The correct answer is 16. An equation of a circle in the xy-plane can be written as (x − t)² + (y − u)² = r², where the center of the circle is (t, u) , the radius of the circle is r, and where t, u, and r are constants. It’s given that the equation of circle A is (x + 5)² + (y − 5)² = 4, which is equivalent to (x + 5)² + (y − 5)² = 2². Therefore, the center of circle A is (−5, 5) and the radius of circle A is 2. It’s given that circle B has the same center as circle A and that the radius of circle B is two times the radius of circle A. Therefore, the center of circle B is (−5, 5) and the radius of circle B is 2 (2), or 4. Substituting −5 for t, 5 for u, and 4 for r into the equation (x − t)² + (y − u)² = r²  yields (x + 5)² + (y − 5)² = 4², which is equivalent to (x + 5)² + (y − 5)² = 16. It follows that the equation of circle B in the xy-plane is (x + 5)² + (y − 5)² = 16. Therefore, the value of k is 16.", hasFigure: false, }, { id: "c8345903", type: "mcq", questionHtml: "The circle above has center O, the length of arc A, D C is 5 π, and x = 100. What is the length of arc A, B C ?", choices: [ { label: "A", text: "9 π" }, { label: "B", text: "13 π" }, { label: "C", text: "18 π" }, { label: "D", text: "13 halves π" }, ], correctAnswer: "B", explanation: "Choice B is correct. The ratio of the lengths of two arcs of a circle is equal to the ratio of the measures of the central angles that subtend the arcs. It’s given that arc A D C is subtended by a central angle with measure 100°. Since the sum of the measures of the angles about a point is 360°, it follows that arc A B C is subtended by a central angle with measure 360 ° − 100 ° = 260 °. If s is the length of arc A B C, then s must satisfy the ratio the fraction s over 5 π, end fraction = the fraction 260 over 100. Reducing the fraction 260 over 100 to its simplest form gives the fraction 13 over 5. Therefore, the fraction s over 5 π, end fraction = the fraction 13 over 5. Multiplying both sides of the fraction s over 5 π, end fraction = the fraction 13 over 5 by 5 π yields s = 13 π.Choice A is incorrect. This is the length of an arc consisting of exactly half of the circle, but arc A B C is greater than half of the circle. Choice C is incorrect. This is the total circumference of the circle. Choice D is incorrect. This is half the length of arc A B C, not its full length.", hasFigure: true, figureUrl: "/practice-images/c8345903_img1.png", }, { id: "ca2235f6", type: "mcq", questionHtml: "A circle has center (expression), and points (expression) and (expression) lie on the circle. The measure of arc (expression) is (expression) and the length of arc (expression) is (expression) inches. What is the circumference, in inches, of the circle?", choices: [ { label: "A", text: "(expression)" }, { label: "B", text: "(expression)" }, { label: "C", text: "(expression)" }, { label: "D", text: "(expression)" }, ], correctAnswer: "D", explanation: "Choice D is correct. It’s given that the measure of arc A B is 45 ° and the length of arc A B is 3 inches. The arc measure of the full circle is 360 °. If x represents the circumference, in inches, of the circle, it follows that (45 °) / (360 °) = (3 inches) / (x inches). This equation is equivalent to (45) / (360) = (3) / (x), or one eighth = (3) / (x). Multiplying both sides of this equation by 8 x yields 1 (x) = 3 (8), or x = 24. Therefore, the circumference of the circle is 24 inches.
Choice A is incorrect. This is the length of arc A B.
Choice B is incorrect and may result from multiplying the length of arc A B by 2.
Choice C is incorrect and may result from squaring the length of arc A B.", hasFigure: false, }, { id: "e80d62c6", type: "mcq", questionHtml: "The equation x² + (y − 2)² = 36 represents circle A. Circle B is obtained by shifting circle A down 4 units in the xy-plane. Which of the following equations represents circle B?", choices: [ { label: "A", text: "x² + (y + 2)² = 36" }, { label: "B", text: "x² + (y − 6)² = 36" }, { label: "C", text: "(x − 4)² + (y − 2)² = 36" }, { label: "D", text: "(x + 4)² + (y − 2)² = 36" }, ], correctAnswer: "A", explanation: "Choice A is correct. The standard form of an equation of a circle in the xy-plane is (x − h)² + (y − k)² = r², where the coordinates of the center of the circle are (h, k) and the length of the radius of the circle is r. The equation of circle A, x² + (y − 2)² = 36, can be rewritten as (x − 0)² + (y − 2)² = 6². Therefore, the center of circle A is at (0, 2) and the length of the radius of circle A is 6. If circle A is shifted down 4 units, the y-coordinate of its center will decrease by 4; the radius of the circle and the x-coordinate of its center will not change. Therefore, the center of circle B is at (0, 2 − 4), or (0 −2), and its radius is 6. Substituting 0 for h, −2 for k, and 6 for r in the equation (x − h)² + (y − k)² = r² yields (x − 0)² + (y − (−2))² = (6)², or x² + (y + 2)² = 36. Therefore, the equation x² + (y + 2)² = 36 represents circle B.
Choice B is incorrect. This equation represents a circle obtained by shifting circle A up, rather than down, 4 units.
Choice C is incorrect. This equation represents a circle obtained by shifting circle A right, rather than down, 4 units.
Choice D is incorrect. This equation represents a circle obtained by shifting circle A left, rather than down, 4 units.", hasFigure: false, }, { id: "ebbf23ae", type: "spr", questionHtml: "A circle in the xy-plane has a diameter with endpoints (2, 4) and (2, 14). An equation of this circle is (x − 2)² + (y − 9)² = r², where r is a positive constant. What is the value of r?", choices: [], correctAnswer: "5", explanation: "The correct answer is 5. The standard form of an equation of a circle in the xy-plane is (x − h)² + (y − k)² = r², where h, k, and r are constants, the coordinates of the center of the circle are (h, k), and the length of the radius of the circle is r. It′s given that an equation of the circle is (x − 2)² + (y − 9)² = r². Therefore, the center of this circle is (2, 9). It’s given that the endpoints of a diameter of the circle are (2, 4) and (2, 14). The length of the radius is the distance from the center of the circle to an endpoint of a diameter of the circle, which can be found using the distance formula, √((x 1 − x 2)² + (y 1 − y 2)²). Substituting the center of the circle (2, 9) and one endpoint of the diameter (2, 4) in this formula gives a distance of √((2 − 2)² + (9 − 4)²), or √(0² + 5²), which is equivalent to 5. Since the distance from the center of the circle to an endpoint of a diameter is 5, the value of r is 5.", hasFigure: false, }, { id: "fb58c0db", type: "spr", questionHtml: "Points A and B lie on a circle with radius 1, and arc A, B has length π over 3. What fraction of the circumference of the circle is the length of arc A, B ?", choices: [], correctAnswer: "", explanation: "The correct answer is one sixth. The circumference, C, of a circle is C = 2 π, r, where r is the length of the radius of the circle. For the given circle with a radius of 1, the circumference is C = 2 π · 1, or C = 2 π. To find what fraction of the circumference the length of arc A, B is, divide the length of the arc by the circumference, which gives the fraction π over 3, end fraction ÷ 2 π. This division can be represented by the fraction π over 3, end fraction · the fraction 1 over 2 π, end fraction = one sixth. Note that 1/6, .1666, .1667, 0.166, and 0.167 are examples of ways to enter a correct answer.", hasFigure: false, }, ];