Mathematics - Chapter 9: Circles

These flashcards cover key concepts and theorems from Chapter 9 on circles, including definitions, properties, and relationships related to angles, chords, and cyclic quadrilaterals.

Angle subtended by a chord

The angle formed at a point outside the chord by joining the endpoints of the chord to that point.

Congruent arcs

Arcs that are equal in length and can be superimposed on each other.

Cyclic quadrilateral

A quadrilateral where all four vertices lie on a circle.

Bisect

To divide something into two equal parts, such as how a perpendicular from the center of a circle bisects a chord.

Equal chords subtend equal angles

A theorem stating that in a circle, equal chords create equal angles at the center.

Theorem 9.1

Equal chords of a circle subtend equal angles at the center.

Perpendicular distance from a point to a line

The shortest distance to a line from a point, measured along a line perpendicular to the given line.

Angle in a semicircle

An angle inscribed in a semicircle that is always a right angle (90°).

Theorem 9.7

The angle subtended by an arc at the center is double the angle subtended by the same arc at any point on the circumference.

Theorem 9.10

The sum of the opposite angles of a cyclic quadrilateral is 180°.

Theorem 9.4

The line drawn through the center of a circle to bisect a chord is perpendicular to that chord.

Circumcircle

The circle that passes through all the vertices of a polygon, particularly a cyclic quadrilateral.

Theorem 9.6

Chords equidistant from the center of a circle are equal in length.

Congruent triangles

Triangles that are identical in shape and size, i.e., their corresponding sides and angles are equal.

Theorem 9.5

Equal chords of a circle (or of congruent circles) are equidistant from the center (or centers).