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Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY Lesson 12: Tangent Segments Classwork Opening Exercise In the diagram, what do you think the length of π§ could be? How do you know? Example In each diagram, try to draw a circle with center π· that is tangent to both rays of β π΅π΄πΆ. a. b. Lesson 12: Tangent Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.80 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY c. In which diagrams did it seem impossible to draw such a circle? Why did it seem impossible? What do you conjecture about circles tangent to both rays of an angle? Why do you think that? Exercises 1. You conjectured that if a circle is tangent to both rays of a circle, then the center lies on the angle bisector. a. Rewrite this conjecture in terms of the notation suggested by the diagram. Given: Need to show: Lesson 12: Tangent Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.81 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY b. 2. Prove your conjecture using a two-column proof. An angle is shown below. a. Draw at least three different circles that are tangent to both rays of the given angle. b. Label the center of one of your circles with π. How does the distance between π and the rays of the angle compare to the radius of the circle? How do you know? Lesson 12: Tangent Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.82 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY 3. Construct as many circles as you can that are tangent to both the given angles at the same time. You can extend the rays as needed. These two angles share a side. C Explain how many circles you can draw to meet the above conditions and how you know. 4. In a triangle, let π be the location where two angle bisectors meet. Must π be on the third angle bisector as well? Explain your reasoning. Lesson 12: Tangent Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.83 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 M5 GEOMETRY 5. Using a straightedge, draw a large triangle π΄π΅πΆ. a. Construct a circle inscribed in the given triangle. b. Explain why your construction works. c. Do you know another name for the intersection of the angle bisectors in relation to the triangle? Lesson 12: Tangent Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.84 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY Lesson Summary THEOREMS: ο§ The two tangent segments to a circle from an exterior point are congruent. ο§ If a circle is tangent to both rays of an angle, then its center lies on the angle bisector. ο§ Every triangle contains an inscribed circle whose center is the intersection of the triangleβs angle bisectors. Problem Set 1. 2. On a piece of paper, draw a circle with center π΄ and a point, πΆ, outside of the circle. a. How many tangents can you draw from πΆ to the circle? b. c. Draw two tangents from πΆ to the circle, and label the tangency points π· and πΈ. Fold your paper along the line π΄πΆ. What do you notice about the lengths of Μ Μ Μ Μ πΆπ· and Μ Μ Μ Μ πΆπΈ ? About the measures of β π·πΆπ΄ and β πΈπΆπ΄? Μ Μ Μ Μ π΄πΆ is the of β π·πΆπΈ. d. Μ Μ Μ Μ πΆπ· and Μ Μ Μ Μ πΆπΈ are tangent to circle π΄. Find π΄πΆ. 2 3 In the figure, the three segments are tangent to the circle at points π΅, πΉ, and πΊ. If π¦ = π₯, find π₯, π¦, and π§. Lesson 12: Tangent Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.85 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY 3. 4. In the figure given, the three segments are tangent to the circle at points π½, πΌ, and π». a. Prove πΊπΉ = πΊπ½ + π»πΉ. b. Find the perimeter of β³ πΊπΆπΉ. In the figure given, the three segments are tangent to the circle at points F, π΅, and πΊ. Find π·πΈ. Lesson 12: Tangent Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.86 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM M5 GEOMETRY 5. 6. β‘ is tangent to circle π΄. If points πΆ and π· are the intersection points of circle π΄ and any line parallel to πΈπΉ β‘ , answer πΈπΉ the following. a. β‘ ? Explain. Does πΆπΊ = πΊπ· for any line parallel to πΈπΉ b. Suppose that β‘πΆπ· coincides with β‘πΈπΉ . Would πΆ, πΊ, and π· all coincide with π΅? c. Suppose πΆ, πΊ, and π· have now reached π΅, so β‘πΆπ· is tangent to the circle. What is the angle between β‘πΆπ· and Μ Μ Μ Μ π΄π΅ ? d. Draw another line tangent to the circle from some point, π, in the exterior of the circle. If the point of tangency is point π, what is the measure of β πππ΄? The segments are tangent to circle π΄ at points π΅ and π·. Μ Μ Μ Μ πΈπ· is a diameter of the circle. Μ Μ Μ Μ Μ Μ Μ Μ β a. Prove π΅πΈ πΆπ΄. b. Prove quadrilateral π΄π΅πΆπ· is a kite. Lesson 12: Tangent Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.87 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 M5 GEOMETRY 7. In the diagram shown, β‘π΅π» is tangent to the circle at point π΅. What is the relationship between β π·π΅π», the angle between the tangent and a chord, and the arc subtended by that chord and its inscribed angle β π·πΆπ΅? Lesson 12: Tangent Segments This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from GEO-M5-TE-1.3.0-10.2015 S.88 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
