Geometry Chapter 9 Review Secant •A line that contains a chord of a circle. . P Tangent •A line in the plane of a circle that intersects the circle in exactly one point. . Point of tangency . P Theorem If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. ● Corollary Tangents to a circle from a point are congruent. ● ● Theorem • In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent. J M 22 ~ LM. If m 1 = m 2, then JK = 7 7 ~ LM, then m 1 = m 2. If JK = 11 m K k 7 7 Theorem In the same circle or in congruent circles: • congruent arcs have congruent chords • congruent chords have congruent arcs ● O Theorem A diameter that is perpendicular to a chord bisects the chord and its arc. ● O Theorem In the same circle or in congruent circles: • chords equally distant from the center (or centers) are congruent • congruent chords are equally distant from the center (or centers) ● O Inscribed Angle Theorem • The measure of the inscribed angle is half the measure of its central angle (and therefore half the intercepted arc). 30o 60o 60o A Very Similar Theorem • The measure of the angle created by a chord and a tangent equals half the intercepted arc. 35o 70o Corollary • If two inscribed angles intercept the same arc, then the angles are congruent. ~ giants sf = x~ =y y x giants sf Corollary • If an inscribed angle intercepts a semicircle, then it is a right angle. Why? 180o 90o Corollary • If a quadrilateral is inscribed in a circle, then opposite angles are supplementary. 70o 85o 95o 110o Interior Angle Theorem The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs. A 1 50˚ D B 60˚ m<1 = ½( mAD + mBC ) C If mAD = 50 and mBC = 60 55 m<1 = ½(50 + 60) = _____ Exterior Angle Theorem The measure of the angles formed by intersecting secants and tangents outside a circle is equal to half the difference of the measures of the intercepted arcs. x˚ y˚ m<1 = ½(x – y) 1 x˚ x˚ y˚ y˚ 2 m<3 = ½(x – y) m<2 = ½(x – y) 3 Theorem • When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the Q S other chord. 4 8 P 8x3=6x4 3 xX 6 R Theorem • When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment. FD x FE = FH x FG E D F G H External x Whole Thing = External x Whole Thing Theorem • When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment. A PB x PC = (PA)2 P C External x Whole Thing = (Tangent)2 B HW: Start After You Finish the Ch.8 Quiz • Chapter 9 W.S.