Parts of a Circle

**Parts of a Circle**

A circle is a set of all points in a plane that are all an equal distance from a single point, the **center**. The distance from a circle's center to a point on the circle is called the radius of the circle. A **radius** is a line segment with one endpoint at the center of the circle and the other endpoint on the circle. For the circle below AD, DB, and DC are radii of a circle with center D.

A line segment that crosses the circle by passing through the center of the circle is called the **diameter.** The diameter twice the length of the radius. In the circle above AC is the diameter of the circle.

A **chord** is a segment that also has endpoints on the circle, but the line does not need to cross through the center. On the circle below BC and AC are chords. A diameter is a chord that passes through the center of the circle.

A **secant** is a line that intersects with a circle at 2 different points. In the circle below, line E is a secant. A **tangent** is a line that intersects with the circle at one point. In the circle below line F is a tangent.

**Tangents**

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

**Example**

Jason is located at a point P, 28 feet from the edge of a circular garden. The distance from Jason to a point of tangency on the garden is 56 feet. What is the radius of the garden?

(*r* + 28) = *r* + 56

*r* + 56*r* + 784 = *r* + 3136

*r* + 56*r* - *r* = 3136 — 784

56*r* = 2352

*r* = 42

**Arcs**

If the measure of angle ADB is less than 180°, then the interior of angle ADB forms the **minor arc** while the exterior forms the **major arc**. The measure of an arc is defined by the measure of its central angle. In the example below, the measure of arc BC is 35.

**Chords of Circles**

In the same circle, two minor arcs are congruent if and only if their corresponding chords are congruent, therefore a point D is called the midpoint and arc PQ arc PR if and only if line PQ line PR.

If the diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc, therefore XZ ZY when arc XW arc WY. If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

**Inscribed Angle**

If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. In the circle below angle QRS = of the measure of arc QS

If a right triangle is inscribed in a circle, then the hypotenuse may be the diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. In the circle below angle Y is a right angle if and only if line XZ is the diameter of the circle

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. In the circle below P, Q, R, and S lie on the circle, with a center at D, if and only if P + R = 180, and Q +S = 180.