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Chapter 10
Circles
| Term | Definition |
|---|---|
| circle | the locus or set of all points in a plane equidistant from a given point called the center of the circle |
| radius | a segment with endpoints at the center and on the circle |
| chord | a segment with endpoints on the circle |
| diameter | a chord that passes through the center and is made up of collinear radii |
| Radius Formula | r= 1/2d |
| Diameter Formula | d=2r |
| congruent circles | two circles are congruent if and only if they have congruent radii |
| concentric circles | coplanar circles that have the same center |
| circumference of a circle | the distance around the circle. |
| Circumference formula | C=2 (pi) r or pi D |
| Sum of Central Angles | The sum of the measures of the central angles of a circle with no interior points in common is 360 |
| Minor Arc | shortest arc connecting two endpoints on a circle. |
| Major Arc | longest arc connecting two endpoints on a circle. |
| Semicircle | an arc with endpoints that lie on a diameter |
| Theorem 10.1 | In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent. |
| Postulate 10.1 | The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. |
| Arc Length | The ratio of the length of an arc (l) to the circumference of the circle is equal to the ratio of the degree measure of the arc is 360. |
| Arc Length Equation | l=x/360 (2pi r) |
| Theorem 10.2 | In the same circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. |
| Theorem 10.3 | If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc. |
| Theorem 10.4 | The perpendicular bisector of a chord is a diameter (or radius) of the circle. |
| Theorem 10.5 | In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center. |
| Inscribed Angle | an angle with a vertex on a circle and sides that contain chords of the circle. |
| Intercepted Arc | An arc with endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle. |
| Theorem 10.6 | If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of the intercepted arc. |
| Theorem 10.7 | If the two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent. |
| Theorem 10.8 | An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle. |
| Theorem 10.9 | If a quadrilateral is inscribed in a circle, then its opposite sides are supplementary. |
| Tangents | a line in the same plane as a circle that intersects the circle in exactly one point, called the point of tangency. |
| Common tangent | a line, ray, or segment that is tangent to two circles in the same plane. |
| Theorem 10.10 | In a plane, a line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency. |
| Theorem 10.11 | If two segments from the same exterior point are tangent to a circle, then they are congruent. |
| Circumscribed Polygons | A polygon is circumscribed about a circle if every side of the polygon is tangent to the circle. |
| Secant | a line that intersects a circle in exactly two points. |
| Theorem 10.12 | If two secants or chords intersect in the interior of a circle, then the measure of an angle formed is one half the sum of the measure of the arcs intercepted by the angle and its vertical angle. |
| Theorem 10.13 | If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one half the measure of its intercepted arc. |
| Theorem 10.14 | 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 one half of the difference of the measures of the intercepted arcs. |
| Vertex of angle is on the circle | one half the measure of the intercepted arc |
| vertex of the angle is inside the circle | one half the measure of the sum of the intercepted arc |
| vertex of the angle is outside the circle | one half the measure of the difference of the intercepted arcs. |
| Theorem 10.15 | If two chords intersect in a circle, then the products of the lengths of the chord segments are equal. |
| Theorem 10.16 | 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 segments. |
| Theorem 10.17 | 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 secant segment. |
| Standard Form, Equation of a Circle | The standard form of the equation of a circle with center (h,k) and radius r is (x-h) squared + (y-k) squared = r squared |
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amgeometry
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