# Copy Material. Geometry Unit 5. Circles With and Without Coordinates. Eureka Math. Eureka Math

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1 Copy Material Geometry Unit 5 Circles With and Without Coordinates Eureka Math Eureka Math

2 Lesson 1 Lesson 1: Thales Theorem Circle A is shown below. 1. Draw two diameters of the circle. 2. Identify the shape defined by the endpoints of the two diameters. 3. Explain why this shape will always result. Lesson 1: Thales Theorem 1

3 Lesson 2 Lesson 2: Circles, Chords, Diameters, and Their Relationships 1. Given circle A shown, AF = AG and BC = 22. Find DE. 2. In the figure, circle P has a radius of 10. AB. DE a. If AB = 8, what is the length of AC? b. If DC = 2, what is the length of AB? Lesson 2: Circles, Chords, Diameters, and Their Relationships 2

4 Lesson 2 Graphic Organizer on Circles Diagram Explanation of Diagram Theorem or Relationship Lesson 2: Circles, Chords, Diameters, and Their Relationships 3

5 Lesson 3 Lesson 3: Rectangles Inscribed in Circles Rectangle ABCD is inscribed in circle P. Boris says that diagonal AC could pass through the center, but it does not have to pass through the center. Is Boris correct? Explain your answer in words, or draw a picture to help you explain your thinking. Lesson 3: Rectangles Inscribed in Circles 4

6 Lesson 4 Lesson 4: Experiments with Inscribed Angles Joey marks two points on a piece of paper, as we did in the Exploratory Challenge, and labels them A and B. Using the trapezoid shown below, he pushes the acute angle through points A and B from below several times so that the sides of the angle touch points A and B, marking the location of the vertex each time. Joey claims that the shape he forms by doing this is the minor arc of a circle and that he can form the major arc by pushing the obtuse angle through points A and B from above. The obtuse angle has the greater measure, so it will form the greater arc, states Joey. Ebony disagrees, saying that Joey has it backwards. The acute angle will trace the major arc, claims Ebony. C C D A B 1. Who is correct, Joey or Ebony? Why? 2. How are the acute and obtuse angles of the trapezoid related? 3. If Joey pushes one of the right angles through the two points, what type of figure is created? How does this relate to the major and minor arcs created above? Lesson 4: Experiments with Inscribed Angles 5

7 Lesson 4 Example 2 Lesson 4: Experiments with Inscribed Angles 6

8 Lesson 5 Lesson 5: Inscribed Angle Theorem and its Application The center of the circle below is O. If angle B has a measure of 15 degrees, find the values of x and y. Explain how you know. Lesson 5: Inscribed Angle Theorem and its Applications 7

9 Lesson 6 Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Find the measure of angles x and y. Explain the relationships and theorems used. Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles 8

10 Lesson 7 Lesson 7: Properties of Arcs 1. Given circle A with diameters BC and DE and mcd = 56. a. a central angle. b. an inscribed angle. c. a chord that is not a diameter. d. What is the measure of CAD? e. What is the measure of CBD? f. 3 angles of equal measure. g. What is the degree measure of CDB? Lesson 7: The Angle Measure of an Arc 9

11 Lesson 8 Lesson 8: Arcs and Chords 1. Given circle A with radius 10, prove BE = DC. 2. Given the circle at right, find mbd. Lesson 8: Arcs and Chords 10

12 Lesson 9 Lesson 9: Arc Length and Areas of Sectors 1. Find the arc length of PQR. 2. Find the area of sector POR. Lesson 9: Arc Length and Areas of Sectors 11

13 Lesson 10 Lesson 10: Unknown Length and Area Problems 1. Given circle A, find the following (round to the nearest hundredth): a. The mbc in degrees. b. The area of sector BC. 2. Find the shaded area (round to the nearest hundredth). 62⁰ Lesson 10: Unknown Length and Area Problems 12

14 Mid-Module Assessment Task 1. Consider a right triangle drawn on a page with sides of lengths 3 cm, 4 cm, and 5 cm. a. Describe a sequence of straightedge and compass constructions that allow you to draw the circle that circumscribes the triangle. Explain why your construction steps successfully accomplish this task. b. What is the distance of the side of the right triangle of length 3 cm from the center of the circle that circumscribes the triangle? c. What is the area of the inscribed circle for the triangle? Module 5: Circles With and Without Coordinates 13

15 Mid-Module Assessment Task 2. A five-pointed star with vertices AA, MM, BB, NN, and CC is inscribed in a circle as shown. Chords AAAA and MMMM intersect at point PP. a. What is the value of mm BBAANN + mm NNMMCC + mm CCCCCC + mm AAAAAA + mm MMCCBB, the sum of the measures of the angles in the points of the star? Explain your answer. b. Suppose MM is the midpoint of the arc AAAA, NN is the midpoint of arc BBBB, and mm BBAANN = 1 mm CCCCCC. What is mm BBPPCC, and why? 2 Module 5: Circles With and Without Coordinates 14

16 Mid-Module Assessment Task 3. Two chords, AAAA and BBBB in a circle with center OO, intersect at right angles at point PP. AAAA equals the radius of the circle. a. What is the measure of the arc AAAA? b. What is the value of the ratio DDDD? Explain how you arrived at your answer. AAAA Module 5: Circles With and Without Coordinates 15

17 Mid-Module Assessment Task 4. a. An arc of a circle has length equal to the diameter of the circle. What is the measure of that arc in radian? Explain your answer. b. Two circles have a common center OO. Two rays from OO intercept the circles at points AA, BB, CC, and DD as shown. Suppose OOOO OOOO = 2 5 and that the area of the sector given by AA, OO, and DD is 10 cm 2. i. What is the ratio of the measure of the arc AAAA to the measure of the arc BBBB? ii. What is the area of the shaded region given by the points AA, BB, CC, and DD? iii. What is the ratio of the length of the arc AAAA to the length of the arc BBBB? Module 5: Circles With and Without Coordinates 16

18 Mid-Module Assessment Task 5. In this diagram, the points PP, QQ, and RR are collinear and are the centers of three congruent circles. QQ is the point of contact of two circles that are externally tangent. The remaining points at which two circles intersect are labeled AA, BB, CC, and DD, as shown. a. Segment AAAA is extended until it meets the circle with center PP at a point XX. Explain, in detail, why XX, PP, and DD are collinear. b. In the diagram, a section is shaded. What percent of the full area of the circle with center QQ is shaded? Module 5: Circles With and Without Coordinates 17

19 Lesson 11 Lesson 11: Properties of Tangents 1. If BC = 9, AB = 6, and AC = 15, is line BC tangent to circle A? Explain. 2. Construct a line tangent to circle A through point B. Lesson 11: Properties of Tangents 18

20 Lesson 12 Lesson 12: Tangent Segments 1. Draw a circle tangent to both rays of this angle. 2. Let B and C be the points of tangency of your circle. Find the measures of ABC and ACB. Explain how you determined your answer. 3. Let P be the center of your circle. Find the measures of the angles in ΔAPB. Lesson 12: Tangent Segments 19

21 Lesson 13 Lesson 13: The Inscribed Angle Alternate a Tangent Angle Find a, b, and c. Lesson 13: The Inscribed Angle Alternate a Tangent Angle 20

22 Lesson 14 Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle 1. Lowell says that m DFC = 1 (123) = 61.5 because it is half of 2 the intercepted arc. Sandra says that you can t determine the measure of DFC because you don t have enough information. Who is correct and why? 2. If m EFC = 99, find and explain how you determined your answer. a. m BFE b. mbe Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle 21

23 Lesson 15 Lesson 15: Secant Angle Theorem, Exterior Case 1. Find x. Explain your answer. 2. Use the diagram to show that mde = y + x and mfg = y x. Justify your work. Lesson 15: Secant Angle Theorem, Exterior Case 22

24 Lesson 16 Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant- Tangent) Diagrams 1. In the circle below, mgf = 30, mde = 120, CG = 6, GH = 2, FH = 3, CF = 4, HE = 9, and FE = 12. a. Find a (m DHE). b. Find b (m DCE) and explain your answer. c. Find x (HD) and explain your answer. d. Find y (DG). Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams 23

25 Lesson 17 Lesson 17: Writing the Equation for a Circle 1. Describe the circle given by the equation (x 7) 2 + (y 8) 2 = Write the equation for a circle with center (0, 4) and radius Write the equation for the circle shown below. 4. A circle has a diameter with endpoints at (6, 5) and (8, 5). Write the equation for the circle. Lesson 17: Writing the Equation for a Circle 24

26 Lesson 18 Lesson 18: Recognizing Equations of Circles 1. The graph of the equation below is a circle. Identify the center and radius of the circle. xx xx + y 2 8y 8 = 0 2. Describe the graph of each equation. Explain how you know what the graph will look like. a. xx 2 + 2xx + y 2 = 1 b. xx 2 + y 2 = 3 c. xx 2 + y 2 + 6xx + 6y = 7 Lesson 18: Recognizing Equations of Circles 25

27 Lesson 19 Lesson 19: Equations for Tangent Lines to Circles Consider the circle (x + 2) 2 + (y 3) 2 = 9. There are two lines tangent to this circle having a slope of Find the coordinates of the two points of tangency. 2. Find the equations of the two tangent lines. Lesson 19: Equations for Tangent Lines to Circles 26

28 Lesson 20 Lesson 20: Cyclic Quadrilaterals 1. What value of x guarantees that the quadrilateral shown in the diagram below is cyclic? Explain. 2. Given quadrilateral GKHJ, m KGJ + m KHJ = 180, m HNJ = 60, KN = 4, NJ = 48, GN = 8, and NH = 24, find the area of quadrilateral GKHJ. Justify your answer. Lesson 20: Cyclic Quadrilaterals 27

29 Lesson 21 Lesson 21: Ptolemy s Theorem What is the length of the chord AC? Explain your answer. Lesson 21: Ptolemy s Theorem 28

30 End-of-Module Assessment Task 1. Let CC be the circle in the coordinate plane that passes though the points (0, 0), (0, 6), and (8, 0). a. What are the coordinates of the center of the circle? b. What is the area of the portion of the interior of the circle that lies in the first quadrant? (Give an exact answer in terms of ππ.) Module 5: Circles With and Without Coordinates 29

31 End-of-Module Assessment Task c. What is the area of the portion of the interior of the circle that lies in the second quadrant? (Give an approximate answer correct to one decimal place.) d. What is the length of the arc of the circle that lies in the first quadrant with endpoints on the axes? (Give an exact answer in terms of ππ.) e. What is the length of the arc of the circle that lies in the second quadrant with endpoints on the axes? (Give an approximate answer correct to one decimal place.) Module 5: Circles With and Without Coordinates 30

32 End-of-Module Assessment Task f. A line of slope 1 is tangent to the circle with point of contact in the first quadrant. What are the coordinates of that point of contact? g. Describe a sequence of transformations that show circle CC is similar to a circle with radius one centered at the origin. h. If the same sequence of transformations is applied to the tangent line described in part (f), will the image of that line also be a line tangent to the circle of radius one centered about the origin? If so, what are the coordinates of the point of contact of this image line and this circle? Module 5: Circles With and Without Coordinates 31

33 End-of-Module Assessment Task 2. In the figure below, the circle with center OO circumscribes AAAAAA. Points AA, BB, and PP are collinear, and the line through PP and CC is tangent to the circle at CC. The center of the circle lies inside AAAAAA. a. Find two angles in the diagram that are congruent, and explain why they are congruent. b. If BB is the midpoint of AAAA and PPPP = 7, what is PPPP? Module 5: Circles With and Without Coordinates 32

34 End-of-Module Assessment Task c. If mm BBBBBB = 50, and the measure of the arc AAAA is 130, what is mm PP? 3. The circumscribing circle and the inscribed circle of a triangle have the same center. a. By drawing three radii of the circumscribing circle, explain why the triangle must be equiangular and, hence, equilateral. Module 5: Circles With and Without Coordinates 33

35 End-of-Module Assessment Task b. Prove again that the triangle must be equilateral, but this time by drawing three radii of the inscribed circle. c. Describe a sequence of straightedge and compass constructions that allows you to draw a circle inscribed in a given equilateral triangle. Module 5: Circles With and Without Coordinates 34

36 End-of-Module Assessment Task 4. a. Show that (xx 2)(xx 6) + (yy 5)(yy + 11) = 0 is the equation of a circle. What is the center of this circle? What is the radius of this circle? b. A circle has diameter with endpoints (aa, bb) and (cc, dd). Show that the equation of this circle can be written as (xx aa)(xx bb) + (yy cc)(yy dd) = 0. Module 5: Circles With and Without Coordinates 35

37 End-of-Module Assessment Task 5. Prove that opposite angles of a cyclic quadrilateral are supplementary. Module 5: Circles With and Without Coordinates 36

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