Eureka Math. Geometry, Module 5. Student File_B. Contains Exit Ticket and Assessment Materials
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1 A Story of Functions Eureka Math Geometry, Module 5 Student File_B Contains and Assessment Materials Published by the non-profit Great Minds. Copyright 2015 Great Minds. No part of this work may be reproduced, sold, or commercialized, in whole or in part, without written permission from Great Minds. Non-commercial use is licensed pursuant to a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license; for more information, go to Great Minds and Eureka Math are registered trademarks of Great Minds. Printed in the U.S.A. This book may be purchased from the publisher at eureka-math.org
2 Packet
3 Lesson 1 Lesson 1: Thales Theorem Circle AA 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 is always the result. Lesson 1: Thales Theorem 1 GEO--ETP
4 Lesson 2 Lesson 2: Circles, Chords, Diameters, and Their Relationships 1. Given circle AA shown, AAAA = AAAA and BBBB = 22. Find DDDD. 2. In the figure, circle PP has a radius of 10. AAAA. DDDD a. If AAAA = 8, what is the length of AAAA? b. If DDDD = 2, what is the length of? AAAA Lesson 2: Circles, Chords, Diameters, and Their Relationships 2 GEO--ETP
5 Lesson 3 Lesson 3: Rectangles Inscribed in Circles Rectangle AAAAAAAA is inscribed in circle PP. Boris says that diagonal AAAA 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 3 GEO--ETP
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 AA and BB. Using the trapezoid shown below, he pushes the acute angle through points AA and BB from below several times so that the sides of the angle touch points AA and BB, 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 AA and BB 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. 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 4 GEO--ETP
7 Lesson 5 Lesson 5: Inscribed Angle Theorem and Its Applications The center of the circle below is OO. If angle BB has a measure of 15 degrees, find the values of xx and yy. Explain how you know. Lesson 5: Inscribed Angle Theorem and Its Applications 5 GEO--ETP
8 Lesson 6 Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Find the measure of angles xx and yy. Explain the relationships and theorems used. Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles 6 GEO--ETP
9 Lesson 7 Lesson 7: The Angle Measure of an Arc Given circle AA with diameters BBBB and DDDD and mmcccc = 56. a. a central angle. b. an inscribed angle. c. a chord that is not a diameter. d. What is the measure of CCCCCC? e. What is the measure of CCCCCC? f. 3 angles of equal measure. g. What is the degree measure of CCCCCC? Lesson 7: The Angle Measure of an Arc 7 GEO--ETP
10 Lesson 8 Lesson 8: Arcs and Chords 1. Given circle AA with radius 10, prove BBBB = DDDD. 2. Given the circle at right, find mmbbbb. Lesson 8: Arcs and Chords 8 GEO--ETP
11 Lesson 9 Lesson 9: Arc Length and Areas of Sectors 1. Find the arc length of PPPPPP. 2. Find the area of sector PPPPPP. Lesson 9: Arc Length and Areas of Sectors 9 GEO--ETP
12 Lesson 10 Lesson 10: Unknown Length and Area Problems 1. Given circle AA, find the following (round to the nearest hundredth). a. mmbbbb in degrees b. Area of sector BBBBBB 2. Find the shaded area (round to the nearest hundredth). 62 Lesson 10: Unknown Length and Area Problems 10 GEO--ETP
13 Lesson 11 Lesson 11: Properties of Tangents 1. If BBBB = 9, AAAA = 6, and AAAA = 15, is BBBB tangent to circle AA? Explain. 2. Construct a line tangent to circle AA through point BB. Lesson 11: Properties of Tangents 11 GEO--ETP
14 Lesson 12 Lesson 12: Tangent Segments 1. Draw a circle tangent to both rays of this angle. 2. Let BB and CC be the points of tangency of your circle. Find the measures of AAAAAA and AAAAAA. Explain how you determined your answer. 3. Let PP be the center of your circle. Find the measures of the angles in AAAAAA. Lesson 12: Tangent Segments 12 GEO--ETP
15 Lesson 13 Lesson 13: The Inscribed Angle Alternate A Tangent Angle Find aa, bb, and cc. Lesson 13: The Inscribed Angle Alternate A Tangent Angle 13 GEO--ETP
16 Lesson 14 Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle 1. Lowell says that mm DDDDDD = 1 (123) = 61 because it is half of the intercepted arc. Sandra says that you cannot 2 determine the measure of DDDDDD because you do not have enough information. Who is correct and why? 2. If mm EEEEEE = 9, find and explain how you determined your answer. a. mm BBBBBB b. mmbbbb Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle 14 GEO--ETP
17 Lesson 15 Lesson 15: Secant Angle Theorem, Exterior Case 1. Find xx. Explain your answer. 2. Use the diagram to show that mmdddd = yy + xx and mmffff = yy xx. Justify your work. Lesson 15: Secant Angle Theorem, Exterior Case 15 GEO--ETP
18 Lesson 16 Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant- Tangent) Diagrams In the circle below, mmgggg = 30, mmdddd = 120, CCCC = 6, GGGG = 2, FFFF = 3, CCCC = 4, HHHH = 9, and FFFF = 12. a. Find aa (mm DDDDDD). b. Find bb (mm DDDDDD), and explain your answer. c. Find xx (HHHH), and explain your answer. d. Find yy (DDDD). Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams 16 GEO--ETP
19 Lesson 17 Lesson 17: Writing the Equation for a Circle 1. Describe the circle given by the equation (xx 7) 2 + (yy 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 17 GEO--ETP
20 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 + yy 2 8yy 8 = 0 2. Describe the graph of each equation. Explain how you know what the graph will look like. a. xx 2 + 2xx + yy 2 = 1 b. xx 2 + yy 2 = 3 c. xx 2 + yy 2 + 6xx + 6yy = 7 Lesson 18: Recognizing Equations of Circles 18 GEO--ETP
21 Lesson 19 Lesson 19: Equations for Tangent Lines to Circles Consider the circle (xx + 2) 2 + (yy 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 19 GEO--ETP
22 Lesson 20 Lesson 20: Cyclic Quadrilaterals 1. What value of xx guarantees that the quadrilateral shown in the diagram below is cyclic? Explain. 2. Given quadrilateral GGGGGGGG, mm KKKKKK + mm KKKKKK = 180, mm HHHHHH = 60, KKKK = 4, NNNN = 48, GGGG = 8, and NNNN = 24, find the area of quadrilateral GGGGGGGG. Justify your answer. Lesson 20: Cyclic Quadrilaterals 20 GEO--ETP
23 Lesson 21 Lesson 21: Ptolemy s Theorem What is the length of the chord AAAA? Explain your answer. Lesson 21: Ptolemy s Theorem 21 GEO--ETP
24 Assessment Packet
25 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? Module 5: Circles With and Without Coordinates 1 GEO--AP
26 Mid-Module Assessment Task c. What is the area of the inscribed circle for the triangle? Module 5: Circles With and Without Coordinates 2 GEO--AP
27 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 BBBBBB + mm NNNNNN + mm CCCCCC + mm AAAAAA + mm MMMMMM, 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 BBBBBB = 1 2 mm CCCCCC. What is mm BBBBBB, and why? Module 5: Circles With and Without Coordinates 3 GEO--AP
28 Mid-Module Assessment Task 3. Two chords, AAAA and BBBB in a circle with center OO, intersect at right angles at point PP. AAAA is equal to the length of 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 4 GEO--AP
29 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 radians? 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 5 GEO--AP
30 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. 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. Module 5: Circles With and Without Coordinates 6 GEO--AP
31 Mid-Module Assessment Task 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 7 GEO--AP
32 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 8 GEO--AP
33 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 9 GEO--AP
34 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. Module 5: Circles With and Without Coordinates 10 GEO--AP
35 End-of-Module Assessment Task 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? 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. Module 5: Circles With and Without Coordinates 11 GEO--AP
36 End-of-Module Assessment Task b. If BB is the midpoint of AAAA and PPPP = 7, what is the length of? PPPP c. If mm BBBBBB = 50, and the measure of the arc AAAA is 130, what is mm PP? Module 5: Circles With and Without Coordinates 12 GEO--AP
37 End-of-Module Assessment Task 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 13 GEO--AP
38 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 14 GEO--AP
39 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 cc) + (yy bb)(yy dd) = 0. Module 5: Circles With and Without Coordinates 15 GEO--AP
40 End-of-Module Assessment Task 5. Prove that opposite angles of a cyclic quadrilateral are supplementary. Module 5: Circles With and Without Coordinates 16 GEO--AP
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