import React, { useRef, useState, useEffect } from "react"; import { ArrowDown, Check, BookOpen, Target, Layers } from "lucide-react"; import CircleTheoremsWidget from "../../../components/lessons/CircleTheoremsWidget"; import TangentPropertiesWidget from "../../../components/lessons/TangentPropertiesWidget"; import PowerOfPointWidget from "../../../components/lessons/PowerOfPointWidget"; import Quiz from "../../../components/lessons/Quiz"; import { CIRCLE_PROP_QUIZ_DATA } from "../../../utils/constants"; import { Frac } from "../../../components/Math"; interface LessonProps { onFinish?: () => void; } const CirclePropertiesLesson: React.FC = ({ onFinish }) => { const [activeSection, setActiveSection] = useState(0); const sectionsRef = useRef<(HTMLElement | null)[]>([]); const scrollToSection = (index: number) => { setActiveSection(index); sectionsRef.current[index]?.scrollIntoView({ behavior: "smooth", block: "start", }); }; useEffect(() => { const observer = new IntersectionObserver( (entries) => { entries.forEach((entry) => { if (entry.isIntersecting) { const index = sectionsRef.current.indexOf( entry.target as HTMLElement, ); if (index !== -1) setActiveSection(index); } }); }, { rootMargin: "-20% 0px -60% 0px" }, ); sectionsRef.current.forEach((section) => { if (section) observer.observe(section); }); return () => observer.disconnect(); }, []); const SectionMarker = ({ index, title, icon: Icon, }: { index: number; title: string; icon: any; }) => { const isActive = activeSection === index; const isPast = activeSection > index; return ( ); }; return (
{/* Section 1: Central vs Inscribed Angles */}
{ sectionsRef.current[0] = el; }} className="min-h-screen flex flex-col justify-center mb-24 pt-20 lg:pt-0" >

Central vs. Inscribed Angles

Circle angle theorems are among the highest-frequency SAT topics. The core relationship is simple: angles and arcs are linked by a factor of 2.

The Central vs. Inscribed Relationship

Central Angle

Vertex at the center. Degree measure equals the intercepted arc.

∠central = arc°

Example: Central angle = 80° → arc = 80°

Inscribed Angle

Vertex on the circle. Measure is exactly half the intercepted arc.

∠inscribed =

Example: Arc = 120° → inscribed angle = 60°

{/* Key Corollaries */}

Key Corollaries (SAT Favorites)

Thales' Theorem

An inscribed angle that intercepts a{" "} semicircle (its chord is a diameter) is always 90°. If you see a triangle inscribed in a circle where one side is the diameter, the opposite angle is 90°.

Inscribed Angles on the Same Arc

All inscribed angles intercepting the same arc are equal, regardless of where on the circle the vertex sits.

Cyclic Quadrilateral

Opposite angles in a quadrilateral inscribed in a circle sum to 180°. So ∠A + ∠C = 180° and ∠B + ∠D = 180°.

{/* Worked Examples */}

Worked Example 1: Find inscribed angle

A central angle is 110°. An inscribed angle intercepts the same arc. Find the inscribed angle.

Arc = 110° (central angle equals arc)

Inscribed angle = ={" "} 55°

Worked Example 2: Cyclic quadrilateral

Quadrilateral ABCD is inscribed in a circle. ∠A = 75°, ∠B = 85°. Find ∠C and ∠D.

∠C = 180° − 75° ={" "} 105° (opposite to A)

∠D = 180° − 85° ={" "} 95° (opposite to B)

{/* Section 2: Tangents */}
{ sectionsRef.current[1] = el; }} className="min-h-screen flex flex-col justify-center mb-24" >

Tangent Properties

A tangent line touches the circle at exactly one point (the point of tangency). Two critical theorems govern all SAT tangent questions.

Two Fundamental Tangent Theorems

Property 1: Tangent-Radius Perpendicularity

A radius drawn to the point of tangency is always{" "} perpendicular to the tangent line — they form a 90° angle. This creates a right triangle you can use with the Pythagorean theorem.

Worked Example:

External point P is 13 units from center O. Radius = 5. Find tangent length PT.

PT² + r² = PO² (right angle at T)

PT² + 25 = 169

PT = √144 ={" "} 12

Property 2: Two Tangents from One External Point

If two tangent segments are drawn from the same external point, they are equal in length. If PA and PB are both tangents from P, then PA = PB.

Worked Example:

From external point P, tangent PA = 3x + 2 and tangent PB = 5x − 4.

Set equal: 3x + 2 = 5x − 4

6 = 2x → x = 3

PA = PB = 11

{/* SAT Trap */}

SAT Trap: Don't Confuse Tangent Line with Tangent Segment

The "two tangents are equal" rule applies to the{" "} segments from the external point to the points of tangency — not to the full tangent lines extending beyond the circle.

{/* Section 3: Power of a Point */}
{ sectionsRef.current[2] = el; }} className="min-h-screen flex flex-col justify-center mb-24" >

Power of a Point

"Power of a Point" relates segment lengths when lines pass through or near a circle. Two main cases appear on the SAT.

The Two Power-of-a-Point Cases

Case 1: Chord-Chord (Inside)

Two chords intersect inside the circle at point P.

a × b = c × d

a and b are the two segments of one chord; c and d are the two segments of the other.

Case 2: Secant-Secant or Tangent-Secant (Outside)

Two secants, or a tangent and secant, from external point P.

ext₁ × whole₁ = ext₂ × whole₂

For tangent: tangent² = ext × whole (since both segments of the tangent chord are equal).

{/* Worked Examples */}

Worked Example 1: Chord-Chord

Two chords intersect inside. Chord 1 has segments 4 and 9. Chord 2 has segments 6 and x.

4 × 9 = 6 × x

36 = 6x → x = 6

Worked Example 2: Tangent-Secant

From external point P: tangent PT = 6, secant passes through circle with external part = 4 and whole length = x.

PT² = ext × whole

6² = 4 × x

36 = 4x → x = 9

Worked Example 3: Secant-Secant

Two secants from P: first has external 3, whole 12. Second has external 4, whole x.

3 × 12 = 4 × x

36 = 4x → x = 9

{/* Section 4: Quiz */}
{ sectionsRef.current[3] = el; }} className="min-h-screen flex flex-col justify-center" >

Practice Time

{CIRCLE_PROP_QUIZ_DATA.map((quiz) => (
))}

Topic Mastered!

); }; export default CirclePropertiesLesson;