Name Date Period. Notes  Tangents. 1. If a line is a tangent to a circle, then it is to the


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1 Name ate Period Notes  Tangents efinition: tangent is a line in the plane of a circle that intersects the circle in eactly one point. There are 3 Theorems for Tangents. 1. If a line is a tangent to a circle, then it is to the drawn to the. Eample: Find the length of if is a tangent and m = 90. = 8 = 9. In a plane, if a line is to a of a circle at the endpoint on the circle, then the line is a of the circle. Eample: In ircle, is MO a tangent? Why or why not? M O 3. If two from the same point are tangent to a circle, then they are.
2 Eamples: 1. Using Rule #3, which segments are congruent? R T. If R T S and R = 5, what is the perimeter of? S 3. 5 T N ommon tangents of two circles: ommon eternal tangents  ommon internal tangents Inscribed polygon: polygon is inscribed in a circle if each of its vertices lies on a circle. ircumscribed polygon: polygon is circumscribed about a circle if each side of the polygon is tangent to the circle.
3 ornell Notes Name: ate: Main Ideas/Questions pplication Eercises: 1. Find w in degrees. Title of Notes: hords & rcs chord is a. I. Theorems for chords in the same circle or congruent circle. Illustration and Geometric Equation Theorem:. =. OP = 6. What is OQ? Illustration and Geometric Equation Theorem: 3. Find z in degrees. 4. Find RS. Illustration and Geometric Equation Theorem: 5. JK LM. If m KM = 68º, what is m JK in degrees? Illustration and Geometric Equation Theorem: nswers: 1.) 70..) 6 units. 3.) ) ) 85.
4 Practice hords, rcs, ngles, Tangents Geometry 11 and 1 Name: ate: Period: 1. If O = 10, E = 16, RM =. m = 150, OH = 6, F = O O R M E H F E 3. chord of a circle is 10 inches long and is 1 inches from the center of the circle. Find the length of the radius. 4. The diameter of a circle is 0 cm. long and a chord is 16 cm. long. Find the distance between the chord and the center of the circle. 5. m = 6. Find each variable. m = m = m 107 O = Q 49 m 100 = E 7. PO = 8. lfonzo s Pizzeria bakes olive pieces in the outer crust M 1 X 90 8 Q 8 90 Y P N O of its 0 inch (diameter) pizza. There is at least one olive piece per inch of crust. ssuming the pizza is cut into eight pieces, how many olive pieces will you get in one slice of pizza? (HINT: Think R LENGTH per slice. How many inches are in one slice s arc length of crust?) 9. E is a common internal tangent of X and Y. X = 3, TV = 3, and YE = 6. Find E. 10. Find. X T P V 8 10 E Y 6 nswers: 1) 4. ) ) 13 in. 4) 6 cm. 5) 65. 6) 49, 53, 156, ) 4. 8).5π 8. 9) ).
5 Name ate Period Practice  Inscribed ngles In circle R, measure of arc = and measure of arc = 3. Find each measure. is a diameter. 1. measure of arc. m 3. measure of arc 4. m R 5. measure of arc 6. m In Q, m =7 and measure of arc =46. Find each measure. is a diameter. 7. measure of arc 8. measure of arc 9. measure of arc 10. m 11. m 1. m Q Use circle, at right, for questions #130. HE is a diameter. 13. Name the intercepted arc for HT. 14. If m HT=5, find measure of arc H 15. Name the intercepted arc for TH. E H 16. Find m EH. 17. If m HT=5, find m EH. 18. Name an inscribed angle. T U 19. Name a central angle. 0. If m HT=5, find. measure of arc EH 1. If m E = 40º and m = 44º, what is m? E. m FGH = measure of arc IJ =
6 ircles: ngle Measures Geometry 14 Name: ate: Period: Vocabulary: (p. 687, hapter 14) Which of the following in the illustration is a tangent? Which of the following in the illustration is a chord? Which of the following in the illustration is a secant? Inside angle = ½ (arc + arc) Outside angle = ½ (big arc small arc) y 1 = + y ( ) y 1 y ( ) y 1 y ( ) y 1 y ( ) pplication Eercises: Show LL Work! Find the missing angle measures in degrees.. Identify the angle you are trying to find.. Mark both intercepted arcs that lie entirely in the interior of the angle.. Substitute in formula and solve. 1.. Find m QPR If m LMJ = 61º, what is the measure of arc KN? 8 nswers:
7 ircles: Segment Lengths Geometry 14 Name: ate: Period: hords, Secants, Tangents whole*outside = whole*outside whole*outside = tangent pplication Eercises: Show LL Work! Find the missing segment = , M = 5, = 14. Find M. M 13. Review: Find each lettered angle. b = c = d = e = f = g = nswers: or b = 90º; c = 4º; d = 70º; e = 48º; f = 13º; g = 5º.
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9 Equations of ircles Geometry 15 Name: ate: Period: Objective: In this worksheet, you will write the standard equation for circles. eriving the Standard Equation for ircles (15) a. What is the horizontal distance between the center (h, k) and (, y)? b. What is the vertical distance between the center (h, k) and (, y)? c. Write an equation relating the horizontal and vertical distances to the length r. (Hint: Right triangle) The equation of a circle in standard form with center (h, k) and radius r is (  h) + (y  k) = r. Eamples: 1. Write the standard equation of a circle with center (, 3) and a radius of 4. Equation:. Write the standard equation of a circle with center (1, ) and a point (0, 6) r = (Shortcut: what is the distance on a number line?) Equation: Graphing calculator: N PRGM (raw), 9:ircle, ircle(h, k, r). The comma button is above 7. If you get an oval, press ZOOM, 5:ZSquare, go back to the home screen (N MOE) and press ENTER. 3. Write the standard equation of a circle with diameter ; (3, 1) ; (0, 9) enter of ircle = r = Equation of ircle: 1. ( ) + (y + 3) = 16. (1) + (y) = (+ 1.5) + (y5) = 18.5
10 Name ate Period Notes  ompleting the Square Standard Form of a quadratic equation: PreP Geometry a b c + + = 0 When an equation does not contain a perfect square we can create one by applying a process called completing the square. perfect square has a relationship between the coefficient and the middle term (b) and the constant term (c). In the trinomial , the coefficient of the term is 10 and the constant term is 5. Notice that if we take half of 10 and square it we get the constant term 5. oefficient of onstant 1 ( 10 ) = 5 = 5 This relationship is true in general and is the idea behind completing the square. Use that same technique to turn these into perfect squares: EX: (14) = 7 7 = 49 Now we will solve quadratic equations by completing the square. EX:  6 = 40 You must find the term that completes the square on the left side of the equation and then add that term to both sides of the equation. 6 + = (6) = 3 (3) = = dd the number to both sides of the equation ( 3) = 49 Factor the trinomial 3 = + 7 Take the square root of both sides Split into two cases & solve 3 = 7 3 = 7 = 10 = 4 {4, 10} Solution heck by factoring the original problem: 6 40 = 0 ( 10)( + 4) = 0 = 10, = 4
11 EX: y 4y 5 = 0 y 4y = 5 y 4y + 4 = y 4y + 4 = 9 Isolate the constant term by adding 5 to both sides omplete the square & add it to both sides 1 (4) =  () = 4 (y ) = 9 Factor the left side of the equation y = + 3 Take the square root of both sides y = 3 y = 3 ivide into cases y = 5 y = 1 {1, 5} Solution heck solutions by factoring or plugging in solutions to original. Try this one: 3. y + 1y + 11 = 0 Steps to ompleting the Square To solve a + b + c = 0 where a 0 by completing the square, use these steps. Step 1 e sure the squared term has a coefficient of 1. If the coefficient of the squared term is 1, proceed to step. If the coefficient of the squared term is not 1 divide each side of the equation by a. Step Put the equation in the correct form. Rewrite so that terms with variables are on one side of the equals sign and the constant is on the other side. Step 3 Square half the coefficient of the first degree term (b). Step 4 dd the new term (step 3) to each side. Step 5 Factor the perfect square trinomial. One side should now be a perfect square trinomial. Factor it as the square of a binomial. Simplify the other side. Step 6 Solve the equation. pply the square root property to complete the solution.
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