Published by the non-pro๏ฌt Great Minds. Copyright ยฉ 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 http://greatminds.net/maps/math/copyright. โGreat Mindsโ and โEureka Mathโ are registered trademarks of Great Minds. Printed in the U.S.A. This book may be purchased from the publisher at eureka-math.org 10 9 8 7 6 5 4 3 2 1 Eureka Math โข Geometry, Module 5 Student File_B Contains Exit Ticket and Assessment Materials A Story of Functions ยฎ
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Exit Ticket Packet - Amazon Web Servicesย ยท Lesson 19: Equations for Tangent Lines to Circles Exit Ticket Consider the circle (๐ฅ๐ฅ+ 2)2+ (๐ฆ๐ฆโ3)2= 9. There are two lines
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Published by the non-profit Great Minds.
Copyright ยฉ 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 http://greatminds.net/maps/math/copyright. โGreat Mindsโ and โEureka Mathโ are registered trademarks of Great Minds.
Printed in the U.S.A. This book may be purchased from the publisher at eureka-math.org 10 9 8 7 6 5 4 3 2 1
Rectangle ๐ด๐ด๐ด๐ด๐ด๐ด๐ด๐ด is inscribed in circle ๐๐. Boris says that diagonal ๐ด๐ด๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ 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.
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Joey marks two points on a piece of paper, as we did in the Exploratory Challenge, and labels them ๐ด๐ด and ๐ต๐ต. Using the trapezoid shown below, he pushes the acute angle through points ๐ด๐ด and ๐ต๐ต from below several times so that the sides of the angle touch points ๐ด๐ด and ๐ต๐ต, 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 ๐ด๐ด and ๐ต๐ต from above. โThe obtuse angle has the greater measure, so it will form the greater arc,โ states Joey.
Ebony disagrees, saying that Joey has it backwards. โThe acute angle will trace the major arc,โ 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 relateto the major and minor arcs created above?
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Lesson 5: Inscribed Angle Theorem and Its Applications
Name Date
Lesson 5: Inscribed Angle Theorem and Its Applications
Exit Ticket The center of the circle below is ๐๐. If angle ๐ต๐ต has a measure of 15 degrees, find the values of ๐ฅ๐ฅ and ๐ฆ๐ฆ. Explain how you know.
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Exit Ticket 1. If ๐ต๐ต๐ต๐ต = 9, ๐ด๐ด๐ต๐ต = 6, and ๐ด๐ด๐ต๐ต = 15, is ๐ต๐ต๐ต๐ต๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ tangent to circle ๐ด๐ด? Explain.
2. Construct a line tangent to circle ๐ด๐ด through point ๐ต๐ต.
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Exit Ticket 1. Draw a circle tangent to both rays of this angle.
2. Let ๐ต๐ต and ๐ถ๐ถ be the points of tangency of your circle. Find the measures of โ ๐ด๐ด๐ต๐ต๐ถ๐ถ and โ ๐ด๐ด๐ถ๐ถ๐ต๐ต. Explain how you determined your answer.
3. Let ๐๐ be the center of your circle. Find the measures of the angles in โณ ๐ด๐ด๐๐๐ต๐ต.
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Exit Ticket 1. Find ๐ฅ๐ฅ. Explain your answer.
2. Use the diagram to show that ๐๐๐ท๐ท๐ท๐ท๏ฟฝ = ๐ฆ๐ฆยฐ + ๐ฅ๐ฅยฐ and ๐๐๐น๐น๐น๐น๏ฟฝ = ๐ฆ๐ฆยฐ โ ๐ฅ๐ฅยฐ. Justify your work.
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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?
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2. A five-pointed star with vertices ๐ด๐ด, ๐๐, ๐ต๐ต, ๐๐, and ๐ถ๐ถ is inscribed in a circle as shown. Chords ๐ด๐ด๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐๐๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ intersect at point ๐๐.
a. What is the value of ๐๐โ ๐ต๐ต๐ด๐ด๐๐ + ๐๐โ ๐๐๐๐๐ถ๐ถ + ๐๐โ ๐ถ๐ถ๐ต๐ต๐ด๐ด + ๐๐โ ๐ด๐ด๐๐๐๐ + ๐๐โ ๐๐๐ถ๐ถ๐ต๐ต, the sum of the measures of the angles in the points of the star? Explain your answer.
b. Suppose ๐๐ is the midpoint of the arc ๐ด๐ด๐ต๐ต, ๐๐ is the midpoint of arc ๐ต๐ต๐ถ๐ถ, and ๐๐โ ๐ต๐ต๐ด๐ด๐๐ = 12๐๐โ ๐ถ๐ถ๐ต๐ต๐ด๐ด.
What is ๐๐โ ๐ต๐ต๐๐๐ถ๐ถ, and why?
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3. Two chords, ๐ด๐ด๐ถ๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ต๐ต๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ in a circle with center ๐๐, intersect at right angles at point ๐๐. ๐ด๐ด๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ is equal to the length of the radius of the circle.
a. What is the measure of the arc ๐ด๐ด๐ต๐ต?
b. What is the value of the ratio ๐ท๐ท๐ท๐ท๐ด๐ด๐ด๐ด
? Explain how you arrived at your answer.
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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 ๐๐. Two rays from ๐๐ intercept the circles at points ๐ด๐ด, ๐ต๐ต, ๐ถ๐ถ, and ๐ต๐ต as shown. Suppose ๐๐๐ด๐ด โถ ๐๐๐ต๐ต = 2 โถ 5 and that the area of the sector given by ๐ด๐ด, ๐๐, and ๐ต๐ต is 10 cm2.
i. What is the ratio of the measure of the arc ๐ด๐ด๐ต๐ต to the measure of
the arc ๐ต๐ต๐ถ๐ถ?
ii. What is the area of the shaded region given by the points ๐ด๐ด, ๐ต๐ต, ๐ถ๐ถ, and ๐ต๐ต?
iii. What is the ratio of the length of the arc ๐ด๐ด๐ต๐ต to the length of the arc ๐ต๐ต๐ถ๐ถ?
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5. In this diagram, the points ๐๐, ๐๐, and ๐ ๐ are collinear and are the centers of three congruent circles. ๐๐ is the point of contact of two circles that are externally tangent. The remaining points at which two circles intersect are labeled ๐ด๐ด, ๐ต๐ต, ๐ถ๐ถ, and ๐ต๐ต, as shown.
a. ๐ด๐ด๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ is extended until it meets the circle with center ๐๐ at a point ๐๐. Explain, in detail, why ๐๐, ๐๐, and ๐ต๐ต are collinear.
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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 ๐๐.)
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.)
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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 ๐๐ circumscribes โณ ๐ด๐ด๐ด๐ด๐ถ๐ถ. Points ๐ด๐ด, ๐ด๐ด, and ๐๐ are collinear, and the line through ๐๐ and ๐ถ๐ถ is tangent to the circle at ๐ถ๐ถ. The center of the circle lies inside โณ ๐ด๐ด๐ด๐ด๐ถ๐ถ.
a. Find two angles in the diagram that are congruent, and explain why they are congruent.
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(๐ฅ๐ฅ โ 2)(๐ฅ๐ฅ โ 6) + (๐ฆ๐ฆ โ 5)(๐ฆ๐ฆ + 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 (๐๐, ๐๐) and (๐๐,๐๐). Show that the equation of this circle can be written as