Geometry Reference
Circles
Circle geometry uses a small vocabulary of lines and angle relationships. The key distinction is whether the angle is at the center, on the circle, inside the circle, or outside the circle.
Solved Examples
Jump directly to a worked example by circle relationship.
Fact Table
| Object | Meaning | Useful Fact |
|---|---|---|
| Radius | Segment from center to circle. | All radii of the same circle are congruent. |
| Diameter | Chord through the center. | Twice the radius; subtends a right angle on the circle. |
| Chord | Segment with endpoints on the circle. | Congruent chords cut congruent arcs. |
| Secant | Line that cuts a circle twice. | Creates exterior angle and length relationships. |
| Tangent | Line that touches a circle once. | Perpendicular to the radius at the point of tangency. |
| Central angle | Vertex at the center. | Equals its intercepted arc. |
| Inscribed angle | Vertex on the circle. | Half its intercepted arc. |
Content Formulas
Central Angle
$$m\angle=\text{intercepted arc}$$
Inscribed Angle
$$m\angle=\frac12(\text{intercepted arc})$$
Interior Chords
$$m\angle=\frac12(\text{arc}_1+\text{arc}_2)$$
Exterior Secants or Tangents
$$m\angle=\frac12(\text{far arc}-\text{near arc})$$
The lines through a circle are usually radius, diameter, chord, secant, or tangent. An angle bisector is a different object: it cuts an angle into two congruent angles.
Classic Examples
Inscribed Angle
An inscribed angle intercepts an arc of $118^\circ$. Find the angle.
Solution Steps
- Use half the intercepted arc measure.
- State the angle: $59^\circ$.
$$m\angle=\frac12(118^\circ)$$
$$=59^\circ$$
Exterior Angle
Two secants form an exterior angle. The far arc is $160^\circ$ and the near arc is $64^\circ$. Find the angle.
Solution Steps
- Subtract the near arc from the far arc and take half the difference.
- Simplify the half-difference.
- State the angle: $48^\circ$.
$$m\angle=\frac12(160^\circ-64^\circ)$$
$$=\frac12(96^\circ)$$
$$=48^\circ$$
Tangent and Radius
A tangent touches a circle at point $T$, and $OT$ is a radius. Find the angle between the tangent and $OT$.
Solution Steps
- Use the fact that a tangent is perpendicular to the radius at the point of tangency.
$$m\angle=90^\circ$$