Bell work 1 Find the measure of Arc ABC, if Arc AB = 3x, Arc BC = (x + 80º), and __ __ AB BC AB  BC AB = 3x º A B C BC = ( x + 80 º )

Bell work 1 Answer Since, __ __ AB BC, then AB BC, thus AB  BC, then AB  BC, thus 3x = x + 80º 2x = 80º x = 40º = AB & BC Therefore AB + BC = ABC = 240º AB = 3x º BC = (x + 80 º) A C B

Bell work 2 You are standing at point X. Point X is 10 feet from the center of the circular water tank and 8 feet from point Y. Segment XY is tangent to the circle P at point Y. What is the radius, r, of the circular water tank? X r 10 ft 8 ft Y P

Bell work 2 Answer Use the Pythagorean Theorem since segment XY is tangent to circle P at Point Y, then it is perpendicular to the radius, r at point Y. r = 6 ft

Unit 3 : Circles: 10.3 Arcs and Chords Objectives: Students will: 1. Use inscribed angles and properties of inscribed angles to solve problems related to circles

Words for Circles 1. Inscribed Angle 2. Intercepted Arc 3. Inscribed Polygons 4. Circumscribed Circles Are there any words/terms that you are unsure of?

Inscribed Angles Inscribed angle – is an angle whose vertex is on the circle and whose sides contain chords of the circle. INSCRIBED ANGLE A B INTERCEPTED ARC, AB Vertex on the circle

Intercepted Arc Intercepted Arc – is the arc that lies in the interior of the inscribed angle and has endpoints on the angle. INSCRIBED ANGLE A B INTERCEPTED ARC, AB Vertex on the circle

(p. 613) Theorem Measure of the Inscribed Angle The measure of an inscribed angle is equal half of the measure of its intercept arc. Central Angle CENTER P P A B Inscribed angle C m ∕_ ABC = ½ m AC

m ∕_ ABC = ½ mAC = 30 º Example 1 Central Angle 60 º A B Measure of the INTERCEPTED ARC = the measure of the Central Angle AC = 60 º C The measure of the inscribed angle ABC = ½ the measure of the intercepted AC. 30 º

Example 2 T R U Find the measure of the intercepted TU, if the inscribed angle R is a right angle.

Example 2 Answer T R U The measure of the intercepted TU = 180º, if the inscribed angle R is a right angle. TU = 180 º

Example 3 T R TU = 86 º U Find the measure of the inscribed angles Q, R,and S, given that their common intercepted TU = 86º Q S

Example 3 Answers T R TU = 86 º U Angles Q, R, and S = ½ their common intercepted arc TU Since their intercepted Arc TU = 86º, then Angle Q = Angle R = Angle S = 43º Q S

(p.614) Theorem 10.9 T IF ∕_ Q and ∕_ S both intercepted TU, then ∕_ Q  ∕_ S U If two inscribed angles of a circle intercepted the same arc, then the angles are congruent Q S

Inscribed vs. Circumscribed Inscribed polygon – is when all of its vertices lie on the circle and the polygon is inside the circle. The Circle then is circumscribed about the polygon Circumscribed circle – lies on the outside of the inscribed polygon intersecting all the vertices of the polygon.

Inscribed vs. Circumscribed The Circle is circumscribed about the polygon. Circumscribed Circle Inscribed Polygon

(p. 615) Theorem If a right triangle is inscribed in a circle, then the hypotenuse is the diameter of the circle. Hypotenuse = Diameter

(p. 615) Converse of Theorem If one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is a right angle. Diameter = Hypotenuse B The triangle is inscribed in the circle and one of its sides is the diameter Angle B is a right angle and measures 90 º

Example Triangle ABC is inscribed in the circle Segment AC = the diameter of the circle. Angle B = 3x. Find the value of x. B AC 3x º

Answer Since the triangle is inscribed in the circle and one of its sides is the diameter = hypotenuse side, then its opposite angle, Angle B, measures 90ºThus, 3x = 90º x = 30º x = 30º

(p. 615) Theorem A quadrilateral can be inscribed in a circle iff its opposite angles are supplementary. X Y P Z The Quadrilateral WXYZ is inscribed in the circle iff / X + / Z = 180º, and / W + / Y = 180º W

Example A quadrilateral WXYZ is inscribed in circle P, if ∕_ X = 103º and ∕_ Y = 115º, Find the measures of ∕_ X = 103º and ∕_ Y = 115º, Find the measures of ∕_ W = ? and ∕_ Z = ? ∕_ W = ? and ∕_ Z = ? X Y P Z The Quadrilateral WXYZ is inscribed in the circle iff / X + / Z = 180º, and / W + / Y = 180º W 103 º 115 º

Example From Theorem ∕_ W = 180º – 115º = 65º and ∕_ W = 180º – 115º = 65º and ∕_ Z = 180º – 103º = 77º ∕_ Z = 180º – 103º = 77º X Y P Z The Quadrilteral WXYZ is inscribed in the circle iff / X + / Z = 180º, and / W + / Y = 180º W 103 º 115 º

Home work PWS 10.3 A P. 617 (9 -22) all

Journal Write two things about “Inscribed Angles” or “Inscribed Polygons” related to circles from this lesson.