Download Lesson 16: Similar Triangles in Circle-Secant (or Circle

Lesson 16
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
GEOMETRY
Lesson 16: Similar Triangles in Circle-Secant (or Circle-SecantTangent) Diagrams
Classwork
Opening Exercise
Identify the type of angle and the angle/arc relationship, and then find the measure of 𝑥.
a.
b.
c.
d.
Lesson 16:
Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M5-TE-1.3.0-10.2015
S.114
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 16
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
GEOMETRY
Exploratory Challenge 1
Measure the lengths of the chords in centimeters, and record them in the table.
A.
B.
C.
D.
Circle
𝒂 (𝐜𝐦)
𝒃 (𝐜𝐦)
𝒄 (𝐜𝐦)
𝒅 (𝐜𝐦)
Do you notice a relationship?
A
B
C
D
Lesson 16:
Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M5-TE-1.3.0-10.2015
S.115
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 16
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
GEOMETRY
Exploratory Challenge 2
Measure the lengths of the chords in centimeters, and record them in the table.
A.
B.
C.
D.
Circle
𝒂 (𝐜𝐦)
𝒃 (𝐜𝐦)
𝒄 (𝐜𝐦)
𝒅 (𝐜𝐦)
Do you notice a relationship?
A
B
C
D
Lesson 16:
Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M5-TE-1.3.0-10.2015
S.116
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 16
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
GEOMETRY
The Inscribed Angle Theorem and Its Family
Diagram
Lesson 16:
How the two shapes overlap
Relationship between
𝒂, 𝒃, 𝒄, and 𝒅
Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M5-TE-1.3.0-10.2015
S.117
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 16
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
GEOMETRY
Lesson Summary
THEOREMS:

When secant lines intersect inside a circle, use 𝑎 ∙ 𝑏 = 𝑐 ∙ 𝑑.

When secant lines intersect outside of a circle, use 𝑎(𝑎 + 𝑏) = 𝑐(𝑐 + 𝑑).

When a tangent line and a secant line intersect outside of a circle, use 𝑎2 = 𝑏(𝑏 + 𝑐).
Relevant Vocabulary
SECANT TO A CIRCLE: A secant line to a circle is a line that intersects a circle in exactly two points.
Problem Set
1.
Find 𝑥.
2.
Find 𝑥.
3.
𝐷𝐹 < 𝐹𝐵, 𝐷𝐹 ≠ 1, 𝐷𝐹 < 𝐹𝐸, and all values are
integers; prove 𝐷𝐹 = 3.
4.
𝐶𝐸 = 6, 𝐶𝐵 = 9, and 𝐶𝐷 = 18. Show 𝐶𝐹 = 3.
Lesson 16:
Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M5-TE-1.3.0-10.2015
S.118
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 16
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
GEOMETRY
5.
Find 𝑥.
6.
Find 𝑥.
7.
Find 𝑥.
8.
Find 𝑥.
9.
In the circle shown, 𝐷𝐸 = 11, 𝐵𝐶 = 10, and 𝐷𝐹 = 8. Find 𝐹𝐸, 𝐵𝐹, 𝐹𝐶.
Lesson 16:
Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M5-TE-1.3.0-10.2015
S.119
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 16
NYS COMMON CORE MATHEMATICS CURRICULUM
M5
GEOMETRY
̂ = 150°, 𝑚𝐷𝐵
̂ = 30°, 𝑚∠𝐶𝐸𝐹 = 60°, 𝐷𝐹 = 8, 𝐷𝐵 = 4, and 𝐺𝐹 = 12.
10. In the circle shown, 𝑚𝐷𝐵𝐺
a.
Find 𝑚∠𝐺𝐷𝐵.
b.
Prove △ 𝐷𝐵𝐹 ~ △ 𝐸𝐶𝐹.
c.
̅̅̅̅ and 𝐺𝐸
̅̅̅̅ .
Set up a proportion using 𝐶𝐸
d.
Set up an equation with 𝐶𝐸 and 𝐺𝐸 using a theorem
for segment lengths from this section.
Lesson 16:
Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from GEO-M5-TE-1.3.0-10.2015
S.120
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.