For each Circle C, find the value of x. Assume that segments that appear to be tangent are tangent. 1. x = 2. x =

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1 Name: ate: Period: Homework - Tangents For each ircle, find the value of. ssume that segments that appear to be tangent are tangent. 1. =. = ( 5) =. = (Leave as simplified radical!) 3 8 In the figure, and both are tangent to circle P and circle Q. lso, P = 8, Q = 5, and m P = 5. Find the measure of each of the following: 5. m = 13. Q = 6. m PG = 1. G = 7. m GP = 15. = P G F Q 8. G = 16. PG = 9. m Q = 17. GQ = 10. m FG = 18. PQ = 11. m FQ = 19. = 1. m F = 0. PX and PY are tangent to from an eternal point P. HJ = 18 and P = 1. (a) What is the distance from to X? (b) What is the distance from to Y? (c) Find PX. X (d) Find PY. J H Y P 1. The minor arc cut off by two tangents to a circle from an outside point is five-sevenths of the major arc. Find the angle formed by the tangents.

2 . and are radii and is a common eternal tangent. = 5, = 15, = 1 (a) Find. (b) Find. (c) Find FG. F G Find the indicated lengths. 3. ircle P is tangent to each side of. = 0. = 11, and = 1. Let Q = and find. Q P. Given: Tangent circles,, and. = 8, = 13, and = 11 Find: The radii of the three circles. 5. Find the perimeter of right triangle WXY if the radius of the circle is and WY = 0 Y W X Tangent relationships are indicated by the diagram. Find the length indicated. 6. JM = 7.1, JK = 7. OT=9, OK=15, = J K.5 O T M K 8. = 10, = 9. F = F =, = 6 find the perimeter of Δ F

3 nswers to Pg 665 # 1, - evens (omit # 6 and 16), 3, 5, 8; Pg 673 #,, 6, 1, 16-19, 6-7, 30. Pg in 8. No; Yes; = in cm km ll are congruent bout 5. in Pg 673. arcs T GH JN ML, segments T GH JN ML, angles TF HFG JKN MKL cm 7. He doesn t know that the chords are equidistant from the center in Review: 1. a) Find the measure of the arc. b) Find the arc length of arc.. Find the area of the shaded sector in the nd circle. 3. Find the area of the shaded segment in the 3 rd circle.. Find the degree measure of the arc of a sector with area 35 if the area of the circle is If =, what fraction of the circle is shaded? (Hint: Let the =. is the center of the big circle. is the diameter of a little circle and is the diameter of a medium circle. Find the areas in terms of.)

4 Inscribed ngles Notes n angle is inscribed if its verte is on the circle and its sides contain chords of the circle. is an inscribed angle. If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc. intercepts If m = 100 then m =. If m = 70, then m = If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then the angles are congruent. 1 intercepts intercepts so 3 intercepts ; intercepts Since m = m, If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle. N The opposite angles of a quadrilateral inscribed in a circle are supplementary. R G The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. m = 1 m

5 1) Name an inscribed angle. ) Name an arc intercepted by P 3) If m P =, find m J ) Find 5) RS and TU are diameters of. L Find RT and m TRS. T N R K S 6) In, PQ RS and m 1 = 38 and m QR = 8. Find: m T = m = m 3 = m = m PT = P Q 1 U R 3 S T 7) In Z,, m = 9, m Z = 10. Find: m = m = m = m = m = m = m = 8) Quadrilateral QRST is inscribed in. If m T = 95 and m S = 100, find m Q and m R. Q R 9) In Q, is a diameter, m =68 and m =96. Find: m = m = m = Q m = Z T S

6 Practice: 1- ngle Measures only Find the measure of each numbered angle. Name: ate: Period: Given circle T, find the value of T 0 T 50 T 70 In K, m O = 98, m OY =8, m Y =60, and m =38. Find: 37. m 38. m m 0. m 3 K 1 O Y 3 1. m. m 5

7 Name ate Period Solve for. hords, Secants and Tangents = = = = = = = = = = = =

8 Geometry Find angles in circles Name ate Period. is a diameter of circle O. and are tangents of circle O m =76 and m =110. Find: H (a) m = (b) m = (i) m = (j) m F= (c) m O= (d) m O= (e) m O= (k) m = (l) m G= (m) m = O G (f) m = (g) m = (n) m O= (o) m O= F (h) m = (p) m H= 3. is a tangent. m F=85, m HJG=55, m GK=75, m =0, m =16, m =10 Find: G F (a) m F = (i) m HG= (b) m FG = (j) m GF= L (c) m GH = (d) m H = (e) m F= (f) m G= (g) m H= (k) m L= (l) m FI= (m) m KH= (n) m HG= (o) m GF= H K J I (h) m KH= nswers: : (a) 70 (b) 10 (c) 10 (d) 38 (e) 17 (f) 35 (g) 5 (h) 76 (i) 5 (j) 38 (k) 5 (l) 93 (m) 5 (n) 38 (o) 35 (p) 55 3: (a) 10 (b) 0 (c) 70 (d) 80 (e) 33 (f) 8 (g) 0 (h) 0 (i) 35 (j) 0 (k) 8 (l) 105 (m) 0 (n) 7 (o) 15

9 PreP/GT Geometry NOTS Name 1-5 quations of ircles ate Period Sketch the graph of the equation.. Write the equation of the circle graphed. - + y + = 36 ( ) ( ) 3. Write the equation of the line tangent to the circle in question # at the point: a. (-, ) b. (1, -3) c. (0, 0) d. (-1, -7) e. (-7, 1) f. (-8, 0). Finding the equation of the circle given 3 points on the circle. Write the equation of the circle that contains the points X( - 6, - 1), Y( -, 3), and Z(, - 5). ) Write the equations for the perpendicular bisectors for Δ XYZ. Tell what segments you used. ) Find the intersection of the two perpendicular bisectors. This will be the circumcenter (the center of the circumscribed circle). (Hint: solve as a system of equations.) ) etermine the radius of the circle. Write the formula and substitution step. ) Write the equation of the circle through X, Y, and Z.

10 Name ate Period Worksheet - ompleting the Square Standard Form of a quadratic equation: a b c PreP Geometry + + = 0 Find the value of c that makes each trinomial a perfect square. 1) + 1 +c ) c 3) + +c ) c 5) t + 0t+c 6) r 9r+ c 7) a + 1a+c 8) h 0h+ c 9) p p+c 10) t + t+ c 6 5 Find the eact solution for each equation by completing the square. 11) y y 5=0 1) + 13= 0 13) = 0 1) = 0 Let s try completing the square for equations of circles. Rewrite the equation in standard form. Solve for the center of the circle and the radius. Standard quation of a ircle ( - h ) + ( y - k ) = r 15) + y + + 6y 3=0 16) + y + 8 y+ 8 = 0 17) + y 10 1y 0 = 0 18) + y y+ 1= 0

11 ircles Review and Word Problems Name: ate: Period: Show all work! This is due the day of the test. Give eact answers if possible. If not, round to nearest tenth. 1. arl is planning to visit a circular park. The radius of the park is 8 miles. He is looking at a map of the park and sees that the park has five landmarks along its edge. The landmarks are connected by paths of equal length for biking. These paths form a regular pentagon inscribed in the circle. If arl bikes along these paths to visit each landmark, how many miles will he bike?. circle is inscribed in a triangle. The points of tangency form the vertices of a triangle inscribed in the circle. What are the angles of the inscribed triangle? 3. ora is wrapping a ribbon around a cylinder-shaped gift bo. The bo has a diameter of 15 inches and the ribbon is 60 inches long. ora is able to wrap the ribbon all the way around the bo once, and then continue so that the second end of the ribbon passes the first end. What is the central angle formed between the ends of the ribbon? Round your answer to the nearest tenth of a degree.. wheel is rolling down an incline. Twelve evenly spaced diameters form spokes of the wheel. When spoke is vertical, which spoke will be perpendicular to the incline? 5. Vanessa looked through her telescope at a mountainous landscape. The figure shows what she saw. ased on the view, approimately what angle does the side of the mountain that runs from to make with the horizontal? 6. omplete the square to write the equation of the circle in standard form. Then give the location of the center and the radius. + y + 16y 13 = 0

12 7. Francisco is a painter. He places a circular canvas on his -frame easel and carefully centers it. The ape of the easel is 30 and the measure of arc is. What is the measure of arc? 8. geostationary satellite is about 35,800 kilometers above arth. How many arc degrees of the planet are visible to a camera in the satellite? 9. The circle below represents arth. The radius of arth is about 600 km. Find the distance d that a person can see on a clear day from each of the following heights h above arth. Round your answer to the nearest tenth of a kilometer. 10. rcheologists and scientists unearthed part of a circular wall. They made the measurements shown in the figure. ased on the information in the figure, what was the radius of the circle? a) 100 m b) 500 m c) 1 km 11. The figure shows the cross-section of an ale held in place by a triangular sleeve. brake etends from the ape of the triangle. When the brake is etended.5 inches into the sleeve, it comes into contact with the ale. a) What is the diameter of the ale? 1. The diameter of the base of a cylindrical milk tank is 59 in. The length of the tank is 70 in. You estimate that the depth of the milk in the tank is 0 in. Find the number of gallons of milk in the tank to the nearest gallon. (1 gal = 31 in. 3 ) (iagram is not to scale.) b) If the base of the triangular sleeve is 6. inches long, then what is the perimeter of the triangular sleeve? Hint: First find the length of the chord, then the area of the sector, and subtract the area of the triangle.

13 13. Some circular nglish gardens, like the one shown here, have paths in the shape of an inscribed regular star. a) Find the measure of an inscribed angle formed by the star in the garden shown here. 1. The radius of arth s equator is about 3960 miles. a) Write the equation of the equator with the center of arth as the origin. b) Find the length of a 1 arc on the equator to the nearest tenth of a mile. c) 1 arc along the equator is 60 nautical miles long. How many miles are in a nautical mile? Round to the nearest tenth. b) What is the measure of an inscribed angle in a garden with a five-pointed star? d) olumbus planned his trip to the ast by going west. He thought each 1 arc was 5 miles long. He estimated that the trip would take 1 days. Use your answer to part (b) to find a better estimate. 15. Graph a circle that contains a diameter with endpoints (, 3) and (, 5) and then write the standard equation of the circle. 16. In the circle with center, X =, = 7, XT is a secant, X and X are tangents, and H is the perpendicular bisector of. Find each measure. X = X = QX = TX = H H = Find the value of to the nearest hundredth. ssume that segments that appear tangent are tangent

14 1. Find each angle measure. SHOW LL WORK! (for eample, write 100+0, 70/, , etc.). Find each angle measure. SHOW LL WORK! T 3. In right triangle, is an altitude. The circles centered at P and Q are inscribed in triangles and, respectively. For = 15 and = 0, compute PQ. Q P nswers: miles.. 50, 60, spoke (-) + (y+8) = 81; center (, -8); radius a) 35.8 km, b) 80.0 km, c) km ft. 11. a) 3.9 in. b) 0.8 in gal. 13. a) 77.1, b) a) + y = 15,681,600. b) 69.1 mi. c) 1. mi. d) about 3 days. 15. ( 1) + (y 1) = X =, X = 5, QX = 18, TX = 3, H = m 1 = 70, m = 35, m 3 = 55, m = 30, m 5 = 55, m 6 = 50, m 7 = 75, m 8 = 5,, m 9 = 80, m 10 = 5.. m 1 = 5, m =, m 3 = 7, m = 0, m 5 =, m 6 = 0, m 7 = 66, m 8 = 56, m 9 = 65, m 10 = 9, m 11 = 35, m 1 = 65, m 13 = 31, m 1 =, m 15 =