Chapter 10 Circles – 5 10 – 6

10.1 Exploring Circles Power Standard #7 Use formulas to determine measurements of two and three dimensional figures (1.2.5)

Circle Definition – Set of all points in a plane equidistant from a given point. Exterior Interior

P

Center Radius Chord Diameter Tangent Secant Segments in Circles Point of Tangency Segment with endpoints being the center and any point on the circle Segment with both endpoints on the circleChord through the center of the circle Line or portion thereof which intersects a circle at two distinct points Line or portion thereof which intersects a circle at one point

Common Internal Tangent Common External Tangent

Concentric Circles

Internally Tangent Circles Externally Tangent Circles 482/1 – 12, 16, 18, 23, , skip 36

10.1 Even Answers

10.2 Properties of Tangents Power Standard #7 Use formulas to determine measurements of two and three dimensional figures (1.2.5)

10 – 1 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

10 – 2 In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle

Which segment is a tangent? C A B D

10 – 3 If two segments from the same exterior point are tangent to the circle, then they are congruent.

A circle inscribed in a polygon – each side of polygon is tangent to the circle

A circle circumscribed about a polygon – each vertex of polygon lies on the circle

Ex. Find the perimeter of the quadrilateral ( ) = 48

53 17 r r What is the radius of the circle? **Diagram is not to scale 488/1 – 10,11 – 25 odd, 33 – 36

10.2 Even Answers

10.3 Central Angles & Arcs Power Standard #7 Use formulas to determine measurements of two and three dimensional figures (1.2.5)

A B C D

A B C D 35 o The measure of a minor arc = the measure of its central angle The measure of a major arc is the difference of 360 and the measure of its associated central angle. E

A B C mAB = m  ACB = 35 o D 35 o 42 o mBD = m  BCD = 42 o mAB + mBD = mAD 35 o + 42 o = 77 o Arc Addition Post. – The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Congruent arcs have same measure and radius

10 – 4 In the same or in congruent circles, two arcs are congruent iff their central angles are congruent. rr xoxo xoxo xoxo A B C D E F AB  CD AB  EF 495/1 – 10, 11 – 29odd, 37 – 40, 45, 47

10.3 Even Answers

10.4 Arcs and Chords Power Standard #7 Use formulas to determine measurements of two and three dimensional figures (1.2.5)

rr A B C D E F 10 – 5 In the same or in congruent circles, two minor arcs are congruent iff their corresponding chords are congruent Arcs and Chords

10 – 6 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc A B C D E O

A C D E O If CD = 12, OE = 8, what is the radius of the circle O? DE = 6 r = r 2 r = = 10

10 – 7 If chord AB is a perpendicular bisector of another chord, then AB is a diameter A B C D E O

r r A B C D E F O P X Y Z 10 – 8 In the same or in congruent circles, two chords are congruent iff they are equidistant from the center. 502/1 – 4, 9 – 18, 26 – 32

10.4 Even Answers

10.5 Inscribed Angles Power Standard #7 Use formulas to determine measurements of two and three dimensional figures (1.2.5)

A B C O Inscribed Angles – Formed by two chords,  ABC

10 – 9 If an angle is inscribed in a circle, then its measure is half the measure of the intercepted arc. A B C O Other case

10 – 10 If two inscribed angles of a circle intercept the same arc, then the angles are congruent A B C O D xoxo xoxo 2x o

10 – 11 An angle that is inscribed in a circle is a right angle iff its intercepted arc is a semicircle.

10 – 12 A quadrilateral can be inscribed in a circle iff its opposite angles are supplementary A B C D W X Y Z xoxo 2x o (360 – 2x) o (180 – x) o x + (180 – x) = 180 o

Which of the following can be inscribed in a circle? 140 o 70 o 508/1 – 14, 15 – 33 odd, 44

10.5 Even Answers

10.6 Other Angle Relationships Power Standard #7 Use formulas to determine measurements of two and three dimensional figures (1.2.5)

10 – 13 If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is half the measure of its intercepted arc. xoxo 2x o yoyo A B 2y o

10 – 14 If two chords intersect in the interior of a circle, then the measure of each angle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 1 2 xoxo yoyo 35 o 81 o 58 o

10 – 15 If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of the intercepted arcs. y x 1

1 xoxo yoyo 25 o 81 o

110 o xoxo (180 – 110) o 70 o

514/1 – 10, 11 – 17 odd, 30, 33, o xoxo (360 – x) o

(h,k) r (x,y) r Standard Equation of a Circle

1. Write the equation of the circle whose center is (4, -9) and it’s radius is 3. r 2 = (x – h) 2 + (y – k) = (x – 4) 2 + (y + 9) 2 2. Write the equation of the circle that has the points (-1, 3) and (1, -1) as endpoints of a diameter of the circle. Sketch the information.What is needed? Center is midpoint Radius is ½ length diameter 5 = (x – 0) 2 + (y – 1) 2 5 = x 2 + (y – 1) 2 HW 522/1 – 22

10 – 9 If an angle is inscribed in a circle, then its measure is half the measure of the intercepted arc. A B C O xoxo xoxo 2x o Return yoyo yoyo 2y o x o + y o 2x o + 2y o = 2(x + y) o D m  ABD=m  ABC + m  CBD mAD = mAC + mCD