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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 Mathโข
Geometry, Module 5
Student File_BContains Exit Ticket
and Assessment Materials
A Story of Functionsยฎ
Exit Ticket Packet
M5 Lesson 1 GEOMETRY
Lesson 1: Thalesโ Theorem
Name Date
Lesson 1: Thalesโ Theorem
Exit Ticket Circle ๐ด๐ด 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.
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M5 Lesson 2 GEOMETRY
Lesson 2: Circles, Chords, Diameters, and Their Relationships
Name Date
Lesson 2: Circles, Chords, Diameters, and Their Relationships
Exit Ticket
1. Given circle ๐ด๐ด shown, ๐ด๐ด๐ด๐ด = ๐ด๐ด๐ด๐ด and ๐ต๐ต๐ต๐ต = 22. Find ๐ท๐ท๐ท๐ท.
2. In the figure, circle ๐๐ has a radius of 10. ๐ด๐ด๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ท๐ท๐ท๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ.
a. If ๐ด๐ด๐ต๐ต = 8, what is the length of ๐ด๐ด๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ?
b. If ๐ท๐ท๐ต๐ต = 2, what is the length of ๐ด๐ด๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ?
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M5 Lesson 3 GEOMETRY
Lesson 3: Rectangles Inscribed in Circles
Name Date
Lesson 3: Rectangles Inscribed in Circles
Exit Ticket
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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M5 Lesson 4 GEOMETRY
Lesson 4: Experiments with Inscribed Angles
Name Date
Lesson 4: Experiments with Inscribed Angles
Exit Ticket
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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M5 Lesson 5 GEOMETRY
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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M5 Lesson 6 GEOMETRY
Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles
Name Date
Lesson 6: Unknown Angle Problems with Inscribed Angles in
Circles
Exit Ticket Find the measure of angles ๐ฅ๐ฅ and ๐ฆ๐ฆ. Explain the relationships and theorems used.
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M5 Lesson 7 GEOMETRY
Lesson 7: The Angle Measure of an Arc
Name Date
Lesson 7: The Angle Measure of an Arc
Exit Ticket
Given circle ๐ด๐ด with diameters ๐ต๐ต๐ต๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ท๐ท๐ท๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐๐๐ต๐ต๐ท๐ท๏ฟฝ = 56ยฐ.
a. Name a central angle.
b. Name an inscribed angle.
c. Name a chord that is not a diameter.
d. What is the measure of โ ๐ต๐ต๐ด๐ด๐ท๐ท?
e. What is the measure of โ ๐ต๐ต๐ต๐ต๐ท๐ท?
f. Name 3 angles of equal measure.
g. What is the degree measure of ๐ต๐ต๐ท๐ท๐ต๐ต๏ฟฝ?
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M5 Lesson 8 GEOMETRY
Lesson 8: Arcs and Chords
Name Date
Lesson 8: Arcs and Chords
Exit Ticket 1. Given circle ๐ด๐ด with radius 10, prove ๐ต๐ต๐ต๐ต = ๐ท๐ท๐ท๐ท.
2. Given the circle at right, find ๐๐๐ต๐ต๐ท๐ท๏ฟฝ .
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M5 Lesson 9 GEOMETRY
Lesson 9: Arc Length and Areas of Sectors
Name Date
Lesson 9: Arc Length and Areas of Sectors
Exit Ticket
1. Find the arc length of ๐๐๐๐๐๐๏ฟฝ .
2. Find the area of sector ๐๐๐๐๐๐.
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M5 Lesson 10 GEOMETRY
Lesson 10: Unknown Length and Area Problems
Name Date
Lesson 10: Unknown Length and Area Problems
Exit Ticket 1. Given circle ๐ด๐ด, find the following (round to the nearest hundredth).
a. ๐๐๐ต๐ต๐ต๐ต๏ฟฝ in degrees
b. Area of sector ๐ต๐ต๐ด๐ด๐ต๐ต
2. Find the shaded area (round to the nearest hundredth).
62ยฐ
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M5 Lesson 11 GEOMETRY
Lesson 11: Properties of Tangents
Name Date
Lesson 11: Properties of Tangents
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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M5 Lesson 12 GEOMETRY
Lesson 12: Tangent Segments
Name Date
Lesson 12: Tangent Segments
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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M5 Lesson 13 GEOMETRY
Lesson 13: The Inscribed Angle AlternateโA Tangent Angle
Name Date
Lesson 13: The Inscribed Angle AlternateโA Tangent Angle
Exit Ticket Find ๐๐, ๐๐, and ๐๐.
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M5 Lesson 14 GEOMETRY
Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle
Name Date
Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle
Exit Ticket
1. Lowell says that ๐๐โ ๐ท๐ท๐ท๐ท๐ท๐ท = 12 (123) = 61ยฐ because it is half of the intercepted arc. Sandra says that you cannot
determine the measure of โ ๐ท๐ท๐ท๐ท๐ท๐ท because you do not have enough information. Who is correct and why?
2. If ๐๐โ ๐ธ๐ธ๐ท๐ท๐ท๐ท = 9ยฐ, find and explain how you determined your answer.
a. ๐๐โ ๐ต๐ต๐ท๐ท๐ธ๐ธ
b. ๐๐๐ต๐ต๐ธ๐ธ๏ฟฝ
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M5 Lesson 15 GEOMETRY
Lesson 15: Secant Angle Theorem, Exterior Case
Name Date
Lesson 15: Secant Angle Theorem, Exterior Case
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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M5 Lesson 16 GEOMETRY
Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams
Name Date
Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-
Tangent) Diagrams
Exit Ticket In the circle below, ๐๐๐บ๐บ๐บ๐บ๏ฟฝ = 30ยฐ, ๐๐๐ท๐ท๐ท๐ท๏ฟฝ = 120ยฐ, ๐ถ๐ถ๐บ๐บ = 6, ๐บ๐บ๐บ๐บ = 2, ๐บ๐บ๐บ๐บ = 3, ๐ถ๐ถ๐บ๐บ = 4, ๐บ๐บ๐ท๐ท = 9, and ๐บ๐บ๐ท๐ท = 12.
a. Find ๐๐ (๐๐โ ๐ท๐ท๐บ๐บ๐ท๐ท).
b. Find ๐๐ (๐๐โ ๐ท๐ท๐ถ๐ถ๐ท๐ท), and explain your answer.
c. Find ๐ฅ๐ฅ (๐บ๐บ๐ท๐ท), and explain your answer.
d. Find ๐ฆ๐ฆ (๐ท๐ท๐บ๐บ).
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M5 Lesson 17 GEOMETRY
Lesson 17: Writing the Equation for a Circle
Name Date
Lesson 17: Writing the Equation for a Circle
Exit Ticket 1. Describe the circle given by the equation (๐ฅ๐ฅ โ 7)2 + (๐ฆ๐ฆ โ 8)2 = 9.
2. Write the equation for a circle with center (0,โ4) and radius 8.
3. 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.
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M5 Lesson 18 GEOMETRY
Lesson 18: Recognizing Equations of Circles
Name Date
Lesson 18: Recognizing Equations of Circles
Exit Ticket 1. The graph of the equation below is a circle. Identify the center and radius of the circle.
๐ฅ๐ฅ2 + 10๐ฅ๐ฅ + ๐ฆ๐ฆ2 โ 8๐ฆ๐ฆ โ 8 = 0
2. Describe the graph of each equation. Explain how you know what the graph will look like.
a. ๐ฅ๐ฅ2 + 2๐ฅ๐ฅ + ๐ฆ๐ฆ2 = โ1
b. ๐ฅ๐ฅ2 + ๐ฆ๐ฆ2 = โ3
c. ๐ฅ๐ฅ2 + ๐ฆ๐ฆ2 + 6๐ฅ๐ฅ + 6๐ฆ๐ฆ = 7
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M5 Lesson 19 GEOMETRY
Lesson 19: Equations for Tangent Lines to Circles
Name Date
Lesson 19: Equations for Tangent Lines to Circles
Exit Ticket Consider the circle (๐ฅ๐ฅ + 2)2 + (๐ฆ๐ฆ โ 3)2 = 9. There are two lines tangent to this circle having a slope of โ1.
1. Find the coordinates of the two points of tangency.
2. Find the equations of the two tangent lines.
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M5 Lesson 20 GEOMETRY
Lesson 20: Cyclic Quadrilaterals
Name Date
Lesson 20: Cyclic Quadrilaterals
Exit Ticket 1. What value of ๐ฅ๐ฅ guarantees that the quadrilateral shown in the diagram below is cyclic? Explain.
2. Given quadrilateral ๐บ๐บ๐บ๐บ๐บ๐บ๐บ๐บ, ๐๐โ ๐บ๐บ๐บ๐บ๐บ๐บ + ๐๐โ ๐บ๐บ๐บ๐บ๐บ๐บ = 180ยฐ, ๐๐โ ๐บ๐บ๐ป๐ป๐บ๐บ = 60ยฐ, ๐บ๐บ๐ป๐ป = 4, ๐ป๐ป๐บ๐บ = 48, ๐บ๐บ๐ป๐ป = 8, and ๐ป๐ป๐บ๐บ = 24, find the area of quadrilateral ๐บ๐บ๐บ๐บ๐บ๐บ๐บ๐บ. Justify your answer.
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M5 Lesson 21 GEOMETRY
Lesson 21: Ptolemyโs Theorem
Name Date
Lesson 21: Ptolemyโs Theorem
Exit Ticket What is the length of the chord ๐ด๐ด๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ? Explain your answer.
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Assessment Packet
Module 5: Circles With and Without Coordinates
M5 Mid-Module Assessment Task GEOMETRY
Name Date
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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Module 5: Circles With and Without Coordinates
M5 Mid-Module Assessment Task GEOMETRY
c. What is the area of the inscribed circle for the triangle?
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Module 5: Circles With and Without Coordinates
M5 Mid-Module Assessment Task GEOMETRY
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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Module 5: Circles With and Without Coordinates
M5 Mid-Module Assessment Task GEOMETRY
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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Module 5: Circles With and Without Coordinates
M5 Mid-Module Assessment Task GEOMETRY
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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Module 5: Circles With and Without Coordinates
M5 Mid-Module Assessment Task GEOMETRY
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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Module 5: Circles With and Without Coordinates
M5 Mid-Module Assessment Task GEOMETRY
b. In the diagram, a section is shaded. What percent of the full area of the circle with center ๐๐ is shaded?
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Module 5: Circles With and Without Coordinates
M5 End-of-Module Assessment Task GEOMETRY
Name Date
1. Let ๐ถ๐ถ 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 ๐๐.)
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Module 5: Circles With and Without Coordinates
M5 End-of-Module Assessment Task GEOMETRY
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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Module 5: Circles With and Without Coordinates
M5 End-of-Module Assessment Task GEOMETRY
f. A line of slope โ1 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 ๐ถ๐ถ is similar to a circle with radius one centered at the origin.
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Module 5: Circles With and Without Coordinates
M5 End-of-Module Assessment Task GEOMETRY
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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Module 5: Circles With and Without Coordinates
M5 End-of-Module Assessment Task GEOMETRY
b. If ๐ด๐ด is the midpoint of ๐ด๐ด๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐๐๐ถ๐ถ = 7, what is the length of ๐๐๐ด๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ?
c. If ๐๐โ ๐ด๐ด๐ด๐ด๐ถ๐ถ = 50ยฐ, and the measure of the arc ๐ด๐ด๐ถ๐ถ is 130ยฐ, what is ๐๐โ ๐๐?
A STORY OF FUNCTIONS
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Module 5: Circles With and Without Coordinates
M5 End-of-Module Assessment Task GEOMETRY
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.
A STORY OF FUNCTIONS
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Module 5: Circles With and Without Coordinates
M5 End-of-Module Assessment Task GEOMETRY
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.
A STORY OF FUNCTIONS
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Module 5: Circles With and Without Coordinates
M5 End-of-Module Assessment Task GEOMETRY
4. a. Show that
(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
(๐ฅ๐ฅ โ ๐๐)(๐ฅ๐ฅ โ ๐๐) + (๐ฆ๐ฆ โ ๐๐)(๐ฆ๐ฆ โ ๐๐) = 0.
A STORY OF FUNCTIONS
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Module 5: Circles With and Without Coordinates
M5 End-of-Module Assessment Task GEOMETRY
5. Prove that opposite angles of a cyclic quadrilateral are supplementary.
A STORY OF FUNCTIONS
ยฉ20 15 G re at Min ds eureka-math.org GEO-M5-AP-1.3.0-09.2015
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