Chapter 10: Circles - Geometry

Chord

segment whose endpoints are on circle

Diameter

chord that contains center of circle

Secant

line that intersects circle in 2 points

Tangent

line in plane of circle that intersects it in exactly 1 point (point of tangency)

Tangents can be

lines, segments, and rays!

Tangent circles

coplanar circles that intersect in 1 point

Concentric circles

coplanar circles that have common center

Common tangent

line/segment that is tangent to 2 coplanar circles

Common internal tangent

intersects segment that joins centers of 2 circles

Common external tangent

doesn’t intersect segment that joins centers of 2 circles

Tangent Line to Circle Theorem

In a plane, line is tangent to circle if and only if line is perpendicular to radius of circle at its endpoint on circle.

External Tangent Congruence Theorem

Tangent segments from common external point are congruent.

Central angle

angle whose vertex is center of circle

Minor arc

measure of angle is <180 degrees

• named by endpoints

Major arc

measure of angle is >180 degrees

• named by endpoints + a point on arc

Semicircle

arc w/ endpoints that are endpoints of diameter

Measure of minor & major arcs

measure of their central angle

Measure of semicircle is

180 degrees

Adjacent arcs

2 arcs of same circle that intersect at exactly 1 point

Arc Addition Postulate

Measure of arc formed by 2 adjacent arcs is sum of measure of the 2 arcs.

Congruent Circles Theorem

2 circles are congruent if and only if they have same radius.

Congruent arcs

if and only if they have same measure and are arcs of same circle/congruent circles

Congruent Central Angles Theorem

In same circle/congruent circles, 2 minor arcs are congruent if and only if corresponding central angles are congruent.

Similar Circles Theorem

All circles are similar.

Similar arcs

if and only if they have same measure

Any chord divides circle into

2 arcs

Diameter divides circle into

2 semicircles

Any other chord divides circle into

a minor arc & major arc

Congruent Corresponding Chords Theorem

In same/congruent circles, 2 minor arcs are congruent if and only if their corresponding chords are congruent.

Perpendicular Chord Bisector Theorem

If diameter of circle is perpendicular to chord, then diameter bisects chord and its arc.

Perpendicular Chord Bisector Converse

If 1 chord of circle is perpendicular bisector of another chord, then 1st chord is a diameter.

Equidistant Chords Theorem

In the same/congruent circles, 2 chords are congruent if and only if they are equidistant from center.

Measure of an Inscribed Angle Theorem

Measure of an inscribed angle is one-half the measure of its intercepted arc.

Inscribed Angles of a Circle Theorem

If 2 inscribed angles of a circle intercept the same arc, then the angles are congruent.

Inscribed Right Triangle Theorem

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if 1 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.

Inscribed Quadrilateral Theorem

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

Tangent and Intersected Chord Theorem

If a tangent and chord intersect at a point on a circle, then the measure of each angle formed is ½ the measure of its intercepted arc.

Angles Inside the Circle Theorem

If 2 chords intersect inside a circle, then the measure of each angle is ½ the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Angles Outside the Circle Theorem

If a tangent and secant, 2 tangents, or 2 secants intersect outside a circle, then the measure of the angle formed is ½ the difference of the measures of the intercepted arcs.

Circumscribed Angle Theorem

The measure of a circumscribed angle is = to 180 degrees minus the measure of the central angle that intercepts the same arc.

Segments of Chords Theorem

If 2 chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is = to the product of the lengths of the segments of the other chord.

Segments of Secants Theorem

If 2 secants segments share the same endpoint outside a circle, then the product of the lengths of 1 secant segment and its external segment is = to the product of the length of the other secant segment and its external segment.

Segments of Secants and Tangents Theorem

If a secant segment and a tangent segments share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment =s the square of the length of the tangent segment.

Standard Equation of a Circle

(x, y) - any point on circle

(h, k) center

radius r

(x-h)²+(y-k)²=r²