Lesson 4: Fundamental Theorem of Similarity (FTS)
Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’3
Lesson 4: Fundamental Theorem of Similarity (FTS) Classwork Exercise In the diagram below, points π and π have been dilated from center π by a scale factor of π = 3.
a.
If the length of |ππ | = 2.3 cm, what is the length of |ππ β² |?
b.
If the length of |ππ| = 3.5 cm, what is the length of |ππ β² |?
Lesson 4: Date:
Fundamental Theorem of Similarity (FTS) 10/30/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’3
c.
Connect the point π to the point π and the point π β² to the point πβ². What do you know about lines π π and π β² π β²?
d.
What is the relationship between the length of segment π π and the length of segment π β² π β²?
e.
Identify pairs of angles that are equal in measure. How do you know they are equal?
Lesson 4: Date:
Fundamental Theorem of Similarity (FTS) 10/30/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.18 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’3
Lesson Summary Theorem: Given a dilation with center π and scale factor π, then for any two points π and π in the plane so that π, π, and π are not collinear, the lines ππ and πβ²πβ² are parallel, where πβ² = π·ππππ‘πππ(π) and πβ² = π·ππππ‘πππ(π), and furthermore, |πβ²πβ²| = π|ππ|.
Problem Set 1.
Use a piece of notebook paper to verify the Fundamental Theorem of Similarity for a scale factor π that is 0 < π < 1. οΌ
Mark a point π on the first line of notebook paper.
οΌ
Mark the point π on a line several lines down from the center π. Draw a ray, βββββ ππ. Mark the point πβ² on the ray, and on a line of the notebook paper, closer to π than you placed point π. This ensures that you have a scale factor that is 0 < π < 1. Write your scale factor at the top of the notebook paper.
οΌ
Draw another ray, ββββββ ππ , and mark the points π and πβ² according to your scale factor.
οΌ
Connect points π and π. Then, connect points πβ² and πβ².
οΌ
Place a point π΄ on line ππ between points π and π. Draw ray βββββ ππ΄. Mark the point π΄β² at the intersection of line βββββ πβ²πβ² and ray ππ΄.
a.
Are lines ππ and πβ²πβ² parallel lines? How do you know?
b.
Which, if any, of the following pairs of angles are equal in measure? Explain.
c.
d.
i.
β πππ and β ππβ²πβ²
ii.
β ππ΄π and β ππ΄β²πβ²
iii.
β ππ΄π and β ππ΄β²πβ²
iv.
β πππ and β ππβ²πβ²
Which, if any, of the following statements are true? Show your work to verify or dispute each statement. i.
|ππβ²| = π|ππ|
ii.
|ππβ²| = π|ππ|
iii.
|πβ²π΄β²| = π|ππ΄|
iv.
|π΄β²πβ²| = π|π΄π|
Do you believe that the Fundamental Theorem of Similarity (FTS) is true even when the scale factor is 0 < π < 1. Explain.
Lesson 4: Date:
Fundamental Theorem of Similarity (FTS) 10/30/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.19 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
3.
8β’3
Caleb sketched the following diagram on graph paper. He dilated points π΅ and πΆ from center π.
a.
What is the scale factor π? Show your work.
b.
Verify the scale factor with a different set of segments.
c.
Which segments are parallel? How do you know?
d.
Which angles are equal in measure? How do you know?
Points π΅ and πΆ were dilated from center π.
a.
What is the scale factor π? Show your work.
b.
If the length of |ππ΅| = 5, what is the length of |ππ΅β² |?
c.
How does the perimeter of triangle ππ΅πΆ compare to the perimeter of triangle ππ΅β²πΆβ²?
d.
Did the perimeter of triangle ππ΅β²πΆβ² = π Γ (perimeter of triangle ππ΅πΆ)? Explain.
Lesson 4: Date:
Fundamental Theorem of Similarity (FTS) 10/30/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.20 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’3
Lesson 4: Fundamental Theorem of Similarity (FTS) Classwork Exercise In the diagram below, points π and π have been dilated from center π by a scale factor of π = 3.
a.
If the length of |ππ | = 2.3 cm, what is the length of |ππ β² |?
b.
If the length of |ππ| = 3.5 cm, what is the length of |ππ β² |?
Lesson 4: Date:
Fundamental Theorem of Similarity (FTS) 10/30/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.17 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’3
c.
Connect the point π to the point π and the point π β² to the point πβ². What do you know about lines π π and π β² π β²?
d.
What is the relationship between the length of segment π π and the length of segment π β² π β²?
e.
Identify pairs of angles that are equal in measure. How do you know they are equal?
Lesson 4: Date:
Fundamental Theorem of Similarity (FTS) 10/30/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.18 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8β’3
Lesson Summary Theorem: Given a dilation with center π and scale factor π, then for any two points π and π in the plane so that π, π, and π are not collinear, the lines ππ and πβ²πβ² are parallel, where πβ² = π·ππππ‘πππ(π) and πβ² = π·ππππ‘πππ(π), and furthermore, |πβ²πβ²| = π|ππ|.
Problem Set 1.
Use a piece of notebook paper to verify the Fundamental Theorem of Similarity for a scale factor π that is 0 < π < 1. οΌ
Mark a point π on the first line of notebook paper.
οΌ
Mark the point π on a line several lines down from the center π. Draw a ray, βββββ ππ. Mark the point πβ² on the ray, and on a line of the notebook paper, closer to π than you placed point π. This ensures that you have a scale factor that is 0 < π < 1. Write your scale factor at the top of the notebook paper.
οΌ
Draw another ray, ββββββ ππ , and mark the points π and πβ² according to your scale factor.
οΌ
Connect points π and π. Then, connect points πβ² and πβ².
οΌ
Place a point π΄ on line ππ between points π and π. Draw ray βββββ ππ΄. Mark the point π΄β² at the intersection of line βββββ πβ²πβ² and ray ππ΄.
a.
Are lines ππ and πβ²πβ² parallel lines? How do you know?
b.
Which, if any, of the following pairs of angles are equal in measure? Explain.
c.
d.
i.
β πππ and β ππβ²πβ²
ii.
β ππ΄π and β ππ΄β²πβ²
iii.
β ππ΄π and β ππ΄β²πβ²
iv.
β πππ and β ππβ²πβ²
Which, if any, of the following statements are true? Show your work to verify or dispute each statement. i.
|ππβ²| = π|ππ|
ii.
|ππβ²| = π|ππ|
iii.
|πβ²π΄β²| = π|ππ΄|
iv.
|π΄β²πβ²| = π|π΄π|
Do you believe that the Fundamental Theorem of Similarity (FTS) is true even when the scale factor is 0 < π < 1. Explain.
Lesson 4: Date:
Fundamental Theorem of Similarity (FTS) 10/30/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.19 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
3.
8β’3
Caleb sketched the following diagram on graph paper. He dilated points π΅ and πΆ from center π.
a.
What is the scale factor π? Show your work.
b.
Verify the scale factor with a different set of segments.
c.
Which segments are parallel? How do you know?
d.
Which angles are equal in measure? How do you know?
Points π΅ and πΆ were dilated from center π.
a.
What is the scale factor π? Show your work.
b.
If the length of |ππ΅| = 5, what is the length of |ππ΅β² |?
c.
How does the perimeter of triangle ππ΅πΆ compare to the perimeter of triangle ππ΅β²πΆβ²?
d.
Did the perimeter of triangle ππ΅β²πΆβ² = π Γ (perimeter of triangle ππ΅πΆ)? Explain.
Lesson 4: Date:
Fundamental Theorem of Similarity (FTS) 10/30/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.20 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
