Circle Theorems

Grade 10 · Math · Study Guide

A comprehensive review of the major theorems governing angles, chords, and tangents in circles.

Circle theorems describe relationships between angles, chords, tangents, and arcs within and around a circle. These theorems are foundational to Euclidean geometry and appear frequently in proofs and problem-solving.

The inscribed angle theorem states that an inscribed angle is half the central angle that subtends the same arc. A direct consequence is that any angle inscribed in a semicircle is exactly 90°, since the central angle for a diameter is 180°.

When two chords intersect inside a circle, the products of their segments are equal: if chords AB and CD intersect at P, then AP × PB = CP × PD. For chords meeting outside the circle (secant-secant or secant-tangent), similar power-of-a-point relationships hold.

Angles formed by tangent-chord pairs equal half the intercepted arc. Two tangents drawn from an external point are equal in length, and the line from the external point to the center bisects the angle between the tangents.

Cyclic quadrilaterals — four-sided figures with all vertices on a circle — have opposite angles that are supplementary (sum to 180°). This property is both necessary and sufficient for a quadrilateral to be cyclic.

Key terms

Inscribed Angle: An angle formed by two chords that share an endpoint on the circle; it equals half the central angle subtending the same arc.
Central Angle: An angle whose vertex is at the center of the circle and whose sides are radii; it equals the intercepted arc in degrees.
Cyclic Quadrilateral: A quadrilateral whose four vertices all lie on a single circle; opposite angles are supplementary.
Power of a Point: For a point P and a circle, the product of signed distances from P along any line through P is constant.
Tangent: A line that touches a circle at exactly one point and is perpendicular to the radius at that point.
Subtend: An arc or chord subtends an angle when that angle's sides pass through the arc's or chord's endpoints.

Flashcards

Inscribed Angle Theorem: An inscribed angle equals half the central angle that intercepts the same arc.
Angle in a Semicircle: Any angle inscribed in a semicircle measures 90°.
Chord Intersection (inside): If two chords intersect inside a circle at P, then AP × PB = CP × PD.
Tangent-Chord Angle: The angle between a tangent and a chord equals half the intercepted arc.
Cyclic Quadrilateral Property: Opposite angles of a cyclic quadrilateral sum to 180°.
Equal Tangents: Two tangents drawn from the same external point to a circle are equal in length.
Tangent-Secant from External Point: tangent² = external segment × whole secant (power of a point).
Perpendicular from Centre to Chord: The perpendicular from the centre of a circle to a chord bisects the chord.

Questions

An inscribed angle intercepts an arc of 140°. What is the measure of the inscribed angle?
1. 70°
2. 140°
3. 280°
4. 35°
Show answer
Answer: 70°
An inscribed angle is half the intercepted arc: 140° ÷ 2 = 70°.
An angle inscribed in a semicircle always measures 90°.
Show answer
Answer: True
A semicircle corresponds to a central angle of 180°, so an inscribed angle in a semicircle is 180° ÷ 2 = 90°.
Two chords PQ and RS intersect inside a circle at point T. If PT = 4, TQ = 9, and RT = 6, what is TS?
1. 6
2. 54
3. 3.5
4. 36
Show answer
Answer: 6
By the intersecting chords theorem: PT × TQ = RT × TS → 4 × 9 = 6 × TS → TS = 36 ÷ 6 = 6.
In a cyclic quadrilateral, opposite angles are ______.
Show answer
Answer: supplementary
A key property of cyclic quadrilaterals is that each pair of opposite angles sums to 180°.
A tangent and a chord meet at a point on a circle. The chord intercepts an arc of 110°. What is the measure of the angle between the tangent and the chord?
1. 55°
2. 110°
3. 70°
4. 125°
Show answer
Answer: 55°
The tangent-chord angle equals half the intercepted arc: 110° ÷ 2 = 55°.

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