Chapter 10: Circles — Arcs, Chords, and Angles
Learning Objectives
- Define circle parts: radius, diameter, chord, arc, central angle, and inscribed angle
- Apply the Central Angle and Arc Measure relationship to compute arc lengths
- Use the Inscribed Angle Theorem and its corollaries to find unknown angles
- Apply the Chord-Chord Power Theorem and the chord-chord angle formula
- Use the Tangent-Radius perpendicularity theorem and equal tangent segment lengths
- Solve problems involving secant-secant and tangent-secant angle measures
10.1 Circle Basics and Central Angles
Definition: Circle and Key Parts
A circle is the set of all points in a plane equidistant from a fixed point called the center. The common distance is the radius $r$.
- Diameter: $d = 2r$ — a chord through the center
- Circumference: $C = 2\pi r = \pi d$
- Arc: a connected portion of the circle. A minor arc is less than a semicircle; a major arc is greater than a semicircle.
- Central angle: an angle whose vertex is the center of the circle
The arc measure (in degrees) equals the measure of its central angle. Arc length converts that degree measure to an actual distance along the circle:
$$\text{Arc Length} = \frac{\theta}{360^\circ} \cdot 2\pi r$$Theorem: Central Angle and Arc Measure
The measure of a central angle equals the measure of its intercepted arc.
$$m\angle AOB = m\overset{\frown}{AB}$$
where $O$ is the center and $A$, $B$ are points on the circle.
Example 10.1 — Arc Length from a Central Angle
A circle has radius $r = 8$. The central angle $\angle AOB = 75°$. Find the arc length of $\overset{\frown}{AB}$.
$$\text{Arc Length} = \frac{75}{360} \cdot 2\pi(8) = \frac{75}{360} \cdot 16\pi \approx \frac{75}{360} \cdot 50.27 \approx \mathbf{10.47}$$
Example 10.2 — Central Angle from Arc Measure
Arc $RS = 140°$. Find the central angle $\angle ROS$.
By the Central Angle and Arc Measure Theorem: $\angle ROS = m\overset{\frown}{RS} = \mathbf{140°}$.
A circle has radius $r = 10$ and an arc with measure $60°$. Find the arc length.
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Circle with center, radius, and 75° central angle — the arc from A to B is highlighted.
Figure 10.1 — Central Angle and Arc on a Circle
10.2 Inscribed Angles and Intercepted Arcs
An inscribed angle has its vertex on the circle, with both sides as chords of the circle. This differs from a central angle, which sits at the center.
Inscribed Angle Theorem and Corollaries
- Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of its intercepted arc. $$m\angle BAC = \frac{1}{2} m\overset{\frown}{BC}$$
- Corollary 1: Inscribed angles that intercept the same arc are congruent.
- Corollary 2: An angle inscribed in a semicircle is a right angle (90°).
- Corollary 3: Opposite angles of an inscribed quadrilateral are supplementary (sum to 180°).
Example 10.3 — Inscribed Angle from Arc Measure
Arc $BC = 100°$. Point $A$ is on the circle (not on arc $BC$). Find inscribed angle $\angle BAC$.
$$\angle BAC = \frac{1}{2}(100°) = \mathbf{50°}$$
Example 10.4 — Inscribed Quadrilateral
Quadrilateral $ABCD$ is inscribed in a circle. $\angle A = 75°$. Find $\angle C$.
Opposite angles of an inscribed quadrilateral are supplementary:
$$\angle C = 180° - \angle A = 180° - 75° = \mathbf{105°}$$
Two inscribed angles $\angle PQR$ and $\angle PSR$ both intercept arc $PR = 84°$. Find both angle measures.
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Key Distinction: A central angle equals its arc. An inscribed angle is half its arc. When the inscribed angle subtends a diameter, the arc is 180° and the angle is 90° — always a right angle.
Circle with inscribed angle at A and central angle at O — both subtend arc BC. Notice the central angle is twice the inscribed angle.
Figure 10.2 — Inscribed Angle vs. Central Angle over the Same Arc
10.3 Chord and Secant Relationships
A chord is a segment whose two endpoints lie on the circle. When two chords intersect inside the circle, they create angle and length relationships.
Chord Intersection Power Theorem
If two chords $\overline{AB}$ and $\overline{CD}$ intersect at point $P$ inside a circle, then:
$$PA \cdot PB = PC \cdot PD$$The products of the two segment lengths are always equal.
Chord-Chord Angle: When two chords intersect at point $P$ inside the circle, the angle formed equals half the sum of the intercepted arcs:
$$\angle P = \frac{1}{2}(m\overset{\frown}{AC} + m\overset{\frown}{BD})$$Secant-Secant (from external point): The angle formed by two secants drawn from an external point equals half the positive difference of the intercepted arcs:
$$\angle P = \frac{1}{2}|m\overset{\frown}{\text{far}} - m\overset{\frown}{\text{near}}|$$Example 10.5 — Chord-Chord Power Theorem
Chords $\overline{AB}$ and $\overline{CD}$ intersect at $P$ inside a circle. $PA = 3$, $PB = 8$, $PC = 4$. Find $PD$.
By the Power Theorem:
$$PA \cdot PB = PC \cdot PD \implies 3 \cdot 8 = 4 \cdot PD \implies PD = \frac{24}{4} = \mathbf{6}$$Example 10.6 — Chord-Chord Angle
Two chords intersect inside a circle. The intercepted arcs are $80°$ and $120°$. Find the chord-chord angle.
$$\angle P = \frac{1}{2}(80° + 120°) = \frac{1}{2}(200°) = \mathbf{100°}$$Chords $\overline{EF}$ and $\overline{GH}$ intersect at $K$ inside a circle. $EK = 5$, $KF = 6$, $GK = 3$. Find $KH$.
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10.4 Tangent Lines and Tangent-Chord Angles
A tangent line touches the circle at exactly one point, called the point of tangency. The tangent is always perpendicular to the radius drawn to that point.
Tangent-Radius Perpendicularity and Tangent Segment Lengths
- Tangent-Radius: A tangent line is perpendicular to the radius at the point of tangency. If $t$ is tangent to circle $O$ at $T$, then $OT \perp t$.
- Tangent-Chord Angle: The measure of an angle formed by a tangent and a chord equals half the intercepted arc. $$\angle = \frac{1}{2} m\overset{\frown}{\text{intercepted arc}}$$
- Equal Tangent Segments: Two tangent segments drawn from the same external point are congruent. $$PA = PB \text{ if } PA \text{ and } PB \text{ are tangents from } P$$
- Pythagorean Relationship: For tangent $t$ from external point $P$ at distance $d$ from center: $$t^2 + r^2 = d^2$$
Example 10.7 — Tangent Length from External Point
External point $P$ is $13$ cm from the center $O$. The radius $r = 5$ cm. Find the length of the tangent from $P$ to the circle.
$$t^2 + r^2 = d^2 \implies t^2 + 5^2 = 13^2 \implies t^2 + 25 = 169 \implies t^2 = 144 \implies t = \mathbf{12 \text{ cm}}$$Example 10.8 — Tangent-Chord Angle
A tangent and a chord form an angle of $65°$ at the point of tangency. Find the measure of the intercepted arc.
Tangent-Chord Angle $= \frac{1}{2}$ intercepted arc, so:
$$\text{Intercepted Arc} = 2 \times 65° = \mathbf{130°}$$Two tangents from external point $P$ touch a circle at $A$ and $B$. $PA = 4x - 3$ and $PB = 2x + 7$. Find $PA$.
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Circle with external point P, tangent line touching at T, dashed radius OT (perpendicular), and dashed hypotenuse OP.
Figure 10.3 — Tangent Line, Radius, and the Right Triangle Relationship
Practice Problems
A circle has radius $r = 6$. Find the arc length of an arc with central angle $120°$.
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Arc $PQ = 200°$. Find the central angle $\angle POQ$ and the inscribed angle $\angle PRQ$ where $R$ is on the major arc.
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Inscribed quadrilateral $ABCD$ has $\angle A = 82°$ and $\angle B = 67°$. Find $\angle C$ and $\angle D$.
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Two chords intersect at $P$. $PA = 6$, $PB = 4$, $PC = 3$. Find $PD$ using the Power Theorem.
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Two chords intersect inside a circle, creating arcs of $50°$ and $90°$ on opposite sides. Find the chord-chord angle.
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External point $P$ is $17$ cm from the center. Radius $= 8$ cm. Find the tangent length from $P$.
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A tangent-chord angle measures $48°$. Find both arcs created by the chord (the intercepted arc and the other arc).
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In a circle, chord $\overline{AB}$ has central angle $100°$. Inscribed angle $\angle ACB$ (with $C$ on the major arc) subtends arc $AB$. A tangent at $A$ forms angle $\theta$ with chord $\overline{AB}$. Find all: $m\overset{\frown}{AB}$, $\angle ACB$, and $\theta$.
Show Solution
$\angle ACB = \frac{1}{2}(100°) = \mathbf{50°}$ (inscribed angle theorem, $C$ on major arc intercepts minor arc).
Tangent-Chord angle at $A$: $\theta = \frac{1}{2}(100°) = \mathbf{50°}$ (tangent-chord angle = half intercepted arc).
Note: $\angle ACB = \theta = 50°$ — an elegant result of circle geometry.
📋 Chapter Summary
Key Formulas
$L = \dfrac{\theta}{360°} \cdot 2\pi r$ where $\theta$ is the central angle in degrees. In radians: $L = r\theta$.
$A = \dfrac{\theta}{360°} \cdot \pi r^2$ — the "pie slice" area corresponding to central angle $\theta$.
An inscribed angle $= \frac{1}{2}$ its intercepted arc. Inscribed angles intercepting the same arc are congruent.
A tangent line is perpendicular to the radius at the point of tangency. $OT \perp$ tangent at $T$.
Angle-Arc Relationships
Central angle $=$ intercepted arc (in degrees). The arc measure equals the central angle that subtends it.
Inscribed angle $= \frac{1}{2}$ intercepted arc. An angle inscribed in a semicircle is always $90°$.
Angle $= \frac{1}{2}(\text{arc}_1 + \text{arc}_2)$. The angle formed by two chords inside a circle averages the two arcs.
Angle $= \frac{1}{2}(\text{larger arc} - \text{smaller arc})$. The angle formed outside is half the difference of the arcs.