Circle Theorems Reference Cheat Sheet

A printable reference covering central angles, inscribed angles, tangent lines, secants, chords, and power of a point for grades 8-10.

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Circle theorems describe how angles, arcs, chords, tangents, and secants are related in and around a circle. Students need this reference because many geometry problems combine several circle rules in one diagram. A clear cheat sheet helps students identify which theorem matches the given information and choose the correct formula quickly. The most important ideas are that central angles match their intercepted arcs, inscribed angles measure half their intercepted arcs, and tangent lines meet radii at right angles. Secant and tangent relationships use products of segment lengths, especially in power of a point problems. When solving, label arcs and segments carefully before substituting into formulas.

Key Facts

A central angle has the same measure as its intercepted arc, so $m\angle AOB = m\widehat{AB}$ .
An inscribed angle measures half its intercepted arc, so $m\angle ACB = \frac{1}{2}m\widehat{AB}$ .
An angle formed by two chords inside a circle equals half the sum of the intercepted arcs, so $m\angle 1 = \frac{1}{2}(m\widehat{AB} + m\widehat{CD})$ .
An exterior angle formed by two secants, two tangents, or a tangent and a secant equals half the difference of the intercepted arcs, so $m\angle 1 = \frac{1}{2}(m\widehat{\text{major}} - m\widehat{\text{minor}})$ .
A tangent line is perpendicular to the radius at the point of tangency, so if $\overline{OT}$ is a radius and $\overline{PT}$ is tangent, then $\angle OTP = 90^\circ$ .
For two intersecting chords inside a circle, the products of the chord segments are equal, so $AE \cdot EB = CE \cdot ED$ .
For two secants from the same external point, the outside segment times the whole secant is equal for both secants, so $PA \cdot PB = PC \cdot PD$ .
For a tangent and a secant from the same external point, the tangent squared equals the outside secant segment times the whole secant, so $PT^2 = PA \cdot PB$ .

Vocabulary

Central angle: An angle whose vertex is at the center of a circle and whose sides are radii.
Inscribed angle: An angle whose vertex lies on the circle and whose sides are chords of the circle.
Tangent: A line that touches a circle at exactly one point, called the point of tangency.
Secant: A line that intersects a circle at exactly two points.
Chord: A segment whose endpoints both lie on the circle.
Power of a point: A relationship showing that products of certain tangent, secant, or chord segment lengths from the same point are equal.

Common Mistakes to Avoid

Using the full arc measure for an inscribed angle is wrong because an inscribed angle is half its intercepted arc, so use $m\angle = \frac{1}{2}m\widehat{\text{arc}}$ .
Adding arcs for an exterior angle is wrong because exterior circle angles use half the difference of the intercepted arcs, so use $m\angle = \frac{1}{2}(m\widehat{\text{larger}} - m\widehat{\text{smaller}})$ .
Using only the outside secant segment in a secant product is wrong because the formula needs outside times whole, so use $PA \cdot PB$ , not just $PA \cdot AB$ unless $AB$ is the whole secant.
Forgetting that a tangent is perpendicular to the radius is wrong because the radius to the point of tangency forms a right angle, so $\angle OTP = 90^\circ$ .
Mixing up chord and secant formulas is wrong because intersecting chords inside the circle use $AE \cdot EB = CE \cdot ED$ , while external secants use outside times whole.

Practice Questions

1 An inscribed angle intercepts an arc measuring $86^\circ$ . What is the measure of the inscribed angle?
2 Two chords intersect inside a circle. If the segments of one chord are $6$ and $10$ , and one segment of the other chord is $8$ , what is the missing segment length?
3 From point $P$ , a tangent has length $12$ and a secant has outside segment length $9$ . What is the whole secant length?
4 A problem gives a tangent, a secant, and a radius drawn to the point of tangency. Which circle theorem should you use first, and why?

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