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
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
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 5. 6. 7. Find the Perimeter! 8.
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!
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 4 3 3 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
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 = 30 2 60º 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.
1. 2. 3. 4. 5. 6. 7) 8) 9) 10) 11) 12)
In circle P shown below, m 1 = 50 and mf = 75. Find each angle and arc below. 9 10 8 7 1 2 P 5 3 F 11 6 4 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 =
ate Period Unit 9 ay 4: Post Holiday reak Review!
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 ( 100 + 40 ) = 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 = ( 110 10 ) = 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
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)
In circle P shown below, T is tangent, E and are diameters, plus me = 50, m = 80, and m = mg. 4 E 15 11 14 7 1 12 P 13 2 5 9 10 8 6 3 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 =
+ 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. 10 10 oth tangents intersect at the same point
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 2 9 4 = 6 = 36 2 9 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
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
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)
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
ate Period You must get at least 10 angles correct to receive any points