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Objectives: To use the relationship between a radius and a tangent To use the relationship between two tangents from one point Tangent to a Circle a line in the plane of the circle that intersects the circle in exactly one point. Point of Tangency the point where a circle and a tangent intersect. A Point of Tangency B Tangent to a Circle Theorem 12.1 ◦ If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency. A O AB ⊥ OP P B Theorem 12.2 ◦ If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. A O P B AB ⊥ ΘO Theorem 12.3 ◦ The two segments tangent to a circle from a point outside the circle are congruent. A B O ~ AB = CB C Homework # 32 Due Tuesday (April 30) Page 665 – 666 ◦# 1 – 20 all Objectives: To use congruent chords, arcs, and central angles To recognize properties of lines through the center of a circle Chord a segment whose endpoints are on a circle P PQ and PQ O Q Theorem 12.4 ◦ Within a circle or in congruent circles: 1. Congruent central angles have congruent chords 2. Congruent chords have congruent arcs 3. Congruent arcs have congruent central angles Theorem 12.5 ◦ Within a circle or in congruent circles: 1. Chords equidistant from the center are congruent 2. Congruent chords are equidistant from the center Theorem 12.6 ◦ In a circle, a diameter that is perpendicular to a chord bisects the chord and its arcs. Theorem 12.7 ◦ In a circle, a diameter that bisects a chord (that is not a diameter) is perpendicular to the chord. Theorem 12.8 ◦ In a circle, the perpendicular bisector of a chord contains the center of the circle. Homework #33 Due Wed/Thurs (May 1/2) Page 673 – 674 # 1 – 19 all Objectives: To find the measure of an inscribed angle To find the measure of an angle formed by a tangent and a chord Inscribed Angle angle in a circle in which the vertex is on the circle and the sides of the angle are chords of the circle Intercepted Arc arc created by drawing an inscribed angle A Intercepted Arc B Inscribed Angle C Theorem 12.9 – Inscribed Angle Theorem ◦ The measure of an inscribed angle is half the measure of its intercepted arc. A m<B = B C 1 mAC 2 Corollaries to the Inscribed Angle Theorem ◦ 1. Two inscribed angles that intercept the same arc are congruent. ◦ 2. An angle inscribed in a semicircle is a right angle ◦ 3. The opposite angles of a quadrilateral inscribed in a circle are supplementary. Theorem 12.10 ◦ The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. m<C = B 1 mBDC 2 B D D C C Homework #34 Due Friday (May 03) Page 681 – 682 ◦# 1 – 22 all Objectives: To find the measures of angles formed by chords, secants, and tangents To find the lengths of segments associated with circles Secant a line that intersects a circle at two points. A O B Theorem 12.11 ◦ The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs. x° 1 y° m<1 = 1 (x 2 + y) Theorem 12.11 cont. ◦ The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs. y° x° 1 m<1 = 1 (x 2 – y) Theorem 12.12 ◦ For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and circle. a x c w P y b d z (w + x)w = (y + z)y a·b=c·d t y z (y + z)y = 𝑡 2 Homework #35 Due Friday (May 03) Page 691 ◦# 1 – 14 all