Geometry Chapter 10

SP and PT are congruent If two segments from the same exterior point are tangent to a circle, then they are congruent.

(Theorem 10-11)

The measure of angle BPC is 1/2(cord BC + cord AD) If two secants or cords intersect in the interior of a circle, then the measure of an angle formed is one half the sum of the measures of the arcs intercepted by the angle & its vertical angle.

(Theorem 10-12)

If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent.

(Theorem 10-7)

An inscribed angle intercepts a diameter or semicircle if and only if the angle is a right angle.

(Theorem 10-8)

B and D are supplementary; C and A are supplementary. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

(Theorem 10-9)

The measure of an arc, formed by two adjacent arcs is the sum of the measures of the two arcs.

Arc Addition Postulate (Postulate 10-1)

Length= (x/360)(2piR)

Arc Length Formula

A segment with points on the circle (all the way inside)

Chord

BA and CD are congruent because they are equidistant from the center. In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

Chord Congruence (Theorem 10-5)

When two cords intersect inside a circle, each chord is divided into two segments called chord segments.

Chord Segments

where (h,k) is the center of the circle and r is the radius, (x-h)^2 + (y-k)^2 = r^2

Circle formula

A line/segment/ray that is tangent to two circles in the same plane.

Common Tangent

Circles that share the same center

Concentric Circles

A line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency.

Determining Tangency (Theorem 10-10)

PT^2 = PQ * PR If a tangent and a secant intersect in the exterior of a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant and its external segment.

Exterior intersection of secant segments and tangents (Theorem 10-17)

If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the angle formed is 1/2 the difference of the measures of the intersected arcs. (The arcs affected by the angle, large angle - small angle)

Exterior intersection of secants & tangents (Theorem 10-14)

A polygon with all of its vertices on the circle. If this happens, the circle is circumscribed and is a circumcircle.

Inscribed

An angle with angle with a vertex on a circle and sides containing chords.

Inscribed angle

The measure of arc AC is twice of the measure of angle ABC If an angle is inscribed in a circle, then the measure of the angle is 1/2 the measure of its intercepted arc.

Inscribed angle + intercepted arc (Theorem 10-6)

Has endpoints in the sides of an inscribed angle and lies on the interior of the inscribed angle.

Intercepted Arc

In the same circle or in congruent circles, two minor arcs are congruent if & only if their central angles are congruent.

Minor Arc Congruence (Theorem 10-1)

In the same circle or in congruent circles, two minor arcs are congruent if & only if their corresponding chords are congruent.

Minor Arc Congruence (Theorem 10-2)

If a diameter or radius of a circle is perpendicular to a chord, then it bisects the chord and its arc.

Perpendicular Bisectors in a Circle (Theorem 10-3)

AB is a diameter of circle O The perpendicular bisector of a chord is a diameter or radius of the circle.

Perpendicular Bisectors in a Circle (pt 2) (Theorem 10-4)

If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is 1/2 the measure of its intercepted arc.

Secant & Tangent Intersection at point of tangency (Theorem 10-13)

PD * PA = PC * PB If two secants intersect in the exterior of a circle, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment.

Secant Segments (Theorem 10-16)

A segment of a secant line that has exactly one endpoint in the circle. A secant segment lying outside the circle is called an external secant segment.

Secant segment

C * D = B * A If two chords intersect in a circle, then the products of the lengths of the chord segments are equal.

Segments of Chords (Theorem 10-15)

A line in the same plane as a circle that intersects the line in exactly one point, called the point of tangency.

Tangent