Topic A: Angles & Arcs in Circles
Lessons 1–5 · Central angles, inscribed angles, and arc relationships.
01–03 — Lessons 1–3
Central & Inscribed Angles
- Central angle: vertex at center. Central angle = intercepted arc.
- Inscribed angle: vertex on circle = half the intercepted arc.
- Semicircle: 180°. Any inscribed angle in a semicircle = 90°.
- Inscribed angles intercepting the same arc are equal.
- Arc addition: adjacent arcs add up.
Central angle = 80° → arc = 80°. Inscribed angle intercepting that arc = 40°.
Practice — Lessons 1–3
Q1 · Multiple Choice
A central angle measures 110°. Its intercepted arc measures:
Q2 · Short Answer
An inscribed angle intercepts an arc of 160°. The inscribed angle measures:
04–05 — Lessons 4–5
Angle Relationships in Circles
- Interior angle (two chords): = (1/2)(arc₁ + arc₂).
- Exterior angle (two secants, secant+tangent, two tangents): = (1/2)|arc₁ − arc₂|.
- Tangent-chord angle: = (1/2)(intercepted arc).
- A tangent is perpendicular to the radius at the point of tangency.
- Two tangents from an external point are congruent.
Two chords intersect inside circle, intercepting arcs 80° and 120°: angle = (80+120)/2 = 100°.
Practice — Lessons 4–5
Q3 · Multiple Choice
Two chords intersect inside a circle, creating arcs of 60° and 100°. The angle formed is:
Q4 · Multiple Choice
A tangent is drawn to a circle. It is perpendicular to:
Topic B: Segments in Circles
Lessons 6–9 · Chord lengths, secants, and tangent-secant relationships.
06–09 — Lessons 6–9
Chord, Secant & Tangent Segment Lengths
- Two chords intersecting inside: a·b = c·d (products of segments are equal).
- Two secants from external point: (whole₁)(external₁) = (whole₂)(external₂).
- Secant + tangent from external point: (whole)(external) = tangent².
- Perpendicular from center to chord: bisects the chord.
- Congruent chords: equidistant from the center.
Chords: segment products 3·8 = 4·x → x = 6.
Practice — Lessons 6–9
Q5 · Multiple Choice
Two chords intersect inside a circle. The segments are 4, 6 and 3, x. Find x using the chord-chord product rule.
Q6 · Multiple Choice
A tangent from external point P has length 9. A secant from P has external segment 3. The whole secant length is:
Topic C: Arc Length, Sector Area & Equations
Lessons 10–14 · Using central angles to find arc length and sector area.
10–12 — Lessons 10–12
Arc Length & Sector Area
- Arc length: L = (θ/360°)·2πr = rθ (θ in radians).
- Sector area: A = (θ/360°)·πr².
- Think of arc length and sector area as 'fractions' of the full circle.
- Full circle: circumference = 2πr; area = πr².
- Segment area: sector area − triangle area.
r=10, central angle=90°: arc length = (90/360)·20π = 5π. Sector area = (1/4)π(100) = 25π.
Practice — Lessons 10–12
Q7 · Short Answer
A circle has radius 6. Central angle = 120°. Find the arc length. (Leave in terms of π.)
Q8 · Multiple Choice
A sector has central angle 45° and radius 8. Its area is:
13–14 — Lessons 13–14
Equation of a Circle
- Standard form: (x − h)² + (y − k)² = r², center (h, k), radius r.
- Rewrite by completing the square when given general form: x² + y² + Cx + Dy + E = 0.
- Points inside the circle: (x−h)² + (y−k)² < r².
- Points outside: (x−h)² + (y−k)² > r².
- On circle: (x−h)² + (y−k)² = r².
Center (3, −2), radius 5: equation (x−3)² + (y+2)² = 25.
Practice — Lessons 13–14
Q9 · Multiple Choice
Which point is on the circle (x−1)² + (y−2)² = 25?
Q10 · Short Answer
Write the equation of a circle centered at (0, 0) with radius 7.