# Unit 9 Syllabus: Circles

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1 ate Period Unit 9 Syllabus: ircles ay Topic 1 Tangent Lines 2 hords and rcs and Inscribed ngles 3 Review/Graded lasswork 4 Review from before reak 5 Finding ngle Measures 6 Finding Segment Lengths 7 Review 8 Test

2 ate Period Tangent Lines - Unit 9 ay 1 1. tangent line is a line that intersects a circle in exactly. The word tangent may refer to a,, or. 2. line that intersects a circle in two points is called a line. a. More on this to come 3. Today s goal: emonstrate an understanding of two important properties of tangents. If a line is drawn tangent to a circle, and a radius is drawn to the point of tangency then the tangent line and radius are. * The converse is also true! Important Property #1 If two lines are tangent to the same circle then the segments from their intersection to the point of tangency are. Important Property #2

3 4. Wonderful applications of the two important properties a. Tangent segments and radii create 5 y 3 5 x 12 3 b. ircumscribed polygons Triangle is about the circle ircle is in the triangle Quadrilateral is in the circle ircle is about the quadrilateral Find the Perimeter! 8.

4 5. pulley system (proof p. 664) 12 x Smaller radius = 10 Larger radius = 24 What is the distance from center to center? a. 48 b. losure: escribe the two properties of tangents that we learned today!

5 ate Period hords, rcs, and Inscribed ngles - Unit 9 ay 2 Important properties that hold within one circle, or in two (or more) congruent circles Z Y X x In Summary a) ongruent central angles have congruent chords b) ongruent chords have congruent arcs c) ongruent arcs have congruent central angles d) hords equidistant from the center are congruent/bisected by the radius

6 Two chords create an angle. The arc that is formed is called an arc. 1. The Inscribed ngle Theorem: The measure of the inscribed angle is the measure of its intercepted arc a) Right-ngle orollary: If an inscribed angle intercepts a semicircle, then the angle is right Y XTZ is a semicircle So XTZ = 180º T O XYZ intercepts XTZ, so it is half the measure Therefore half of 180º is 90º Z b) rc-intercept orollary: If two inscribed angles intercept the same arc, then they have the same measure. intercepts so it is half the measure, 1 Thus = 60 = º intercepts so it is half the measure, 1 Thus = 60 = 30 2 ecause both angles have to be half of 60º - that means that they must both be equal. Thus, m = m because they both equal 30º. c) The opposite angles of a quadrilateral inscribed in a circle are.

7 ) 8) 9) 10) 11) 12)

8 In circle P shown below, m 1 = 50 and mf = 75. Find each angle and arc below P 5 3 F E m 2 = m 9 = m 3 = m 10 = m 4 = m 11 = m 5 = m = m 6 = m = m 7 = m = m 8 = me = mef =

9 ate Period Unit 9 ay 4: Post Holiday reak Review!

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11 ate Period Finding ngle Measures - Unit 9 ay 5 1. The measure of an angle formed by two secants or chords that intersect in the interior of a circle is the of the measures of the arcs intercepted by the angle and its vertical angle. Intersects INSIE the circle. Note: F is NOT the center X F 40º Y ngles are congruent because they are vertical angles XFW = 110 because it is supplementary W 100º Z 1 WFZ = 70 2 ( ) = INSIE the circle is + 2. The measure of an angle formed in the exterior of a circle is the of the measures of the intercepted arcs. 1 MG = ( ) = 50 Intersection OUTSIE the circle 2 110º 10º OUTSIE the circle is - G E M F X R 260º Y 60º Z 20º 15 S 100º T

12 Summary of ngles Vertex is on the circle Vertex is inside the circle Vertex is outside the circle X Y W X F Y Z G F E M 1 X = 2 ( ) 1 Y = 2 ( ) 1 WFZ = + 2 ( WZ XY ) 1 WFX = + 2 ( WX YZ) 1 MG = 2 ( G EF ) In the space below, compare and contrast the three properties shown above, as well as the central angle theorem (how does the measure of a central angle compare to the measure of its arc)? raw pictures if you want irections: Find each missing variable below 1) 2) 3)

13 In circle P shown below, T is tangent, E and are diameters, plus me = 50, m = 80, and m = mg. 4 E P T G m 1 = m 2 = m 9 = m 3 = m 10 = m 4 = m 11 = m 5 = m 12 = m 6 = m 13 = m 7 = m 14 = m 8 = m 15 =

14 + ate Period Segment Lengths - Unit 9 ay 6 1. We can find the lengths of pieces of chords, secants and tangents a. Identifying the different parts of secant and tangent lines. tangent line X is a tangent segment secant line is a chord X X is a secant segment referred to as the Whole X is an external secant segment referred to as the Outside b. Two Tangent Segments Theorem i. If two segments are tangent to a circle at the same external point, then the segments are congruent (we saw this on day 1 already!) because they are both tangent segments oth tangents intersect at the same point

15 c. Two Secants Theorem i. If two secants intersect outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment 18 8 X X and X are both secant segments (Whole) X and X are both external segments (Outside) 16 9 Two secants intersect outside the circle Whole X Outside = Whole X Outside X X = X X 18 8 = 16 9 = 144 d. Secant and Tangent Theorem i. If a secant and a tangent intersect outside a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. EX is a tangent segment X is a secant segment (Whole) GX is an external secant segment (Outside) E 6 G 4 X Whole X Outside = Tangent Squared X GX = EX = 6 = e. Two hords Theorem i. If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segment of the other chord. X is not the center! 3 4 X 8 6 X X = X X 4 6 = 3 8 = 24

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17 Wrap Up - Summary of Segment Lengths irections: Write formulas, in terms of the letters given, that illustrate the 3 formulas a c d b x w y z t z y

18 ate Period Unit 9 Review Unit 9 ay 7 1) 2) Find m Given: is tangent to circle O at, mef = 214, m = 144. Find me 3) 4) Find m 5) 6)

19 7) 8) 9) 10) 100º me m m E m 140º P 40º E PE PE EP E 11) 12) P is the center. Prove: P P O = 5 = 2 3 O =? ( rounded) P

20 ate Period You must get at least 10 angles correct to receive any points

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