Download Theorems Involving Similarity

Theorems Involving Similarity
CK-12
Kaitlyn Spong
Say Thanks to the Authors
Click http://www.ck12.org/saythanks
(No sign in required)
To access a customizable version of this book, as well as other
interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to
reduce the cost of textbook materials for the K-12 market both in
the U.S. and worldwide. Using an open-source, collaborative, and
web-based compilation model, CK-12 pioneers and promotes the
creation and distribution of high-quality, adaptive online textbooks
that can be mixed, modified and printed (i.e., the FlexBook®
textbooks).
Copyright © 2016 CK-12 Foundation, www.ck12.org
The names “CK-12” and “CK12” and associated logos and the
terms “FlexBook®” and “FlexBook Platform®” (collectively
“CK-12 Marks”) are trademarks and service marks of CK-12
Foundation and are protected by federal, state, and international
laws.
Any form of reproduction of this book in any format or medium,
in whole or in sections must include the referral attribution link
http://www.ck12.org/saythanks (placed in a visible location) in
addition to the following terms.
Except as otherwise noted, all CK-12 Content (including CK-12
Curriculum Material) is made available to Users in accordance
with the Creative Commons Attribution-Non-Commercial 3.0
Unported (CC BY-NC 3.0) License (http://creativecommons.org/
licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated
herein by this reference.
Complete terms can be found at http://www.ck12.org/about/
terms-of-use.
Printed: April 25, 2016
AUTHORS
CK-12
Kaitlyn Spong
www.ck12.org
C HAPTER
Chapter 1. Theorems Involving Similarity
1
Theorems Involving
Similarity
Here you will use similar triangles to prove new theorems about triangles.
Can you find any similar triangles in the picture below?
Theorems on Similar Triangles
If two triangles are similar, then their corresponding angles are congruent and their corresponding sides are proportional. There are many theorems about triangles that you can prove using similar triangles.
1. Triangle Proportionality Theorem: A line parallel to one side of a triangle divides the other two sides of the
triangle proportionally. This theorem and its converse will be explored and proved in Example A, Example B,
and the practice exercises.
2. Triangle Angle Bisector Theorem: The angle bisector of one angle of a triangle divides the opposite side of
the triangle into segments proportional to the lengths of the other two sides of the triangle. This theorem will
be explored and proved in Example C.
3. Pythagorean Theorem: For a right triangle with legs a and b and hypotenuse c, a2 + b2 = c2 . This theorem
will be explored and proved in the Guided Practice problems.
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/75403
Solve the following problems
Prove that ∆ADE ∼ ∆ABC.
1
www.ck12.org
The two triangles share 6 A. Because DEkBC, corresponding angles are congruent. Therefore, 6 ADE ∼
= 6 ABC. The
two triangles have two pairs of congruent angles. Therefore, ∆ADE ∼ ∆ABC by AA ∼.
Use your result from Example A to prove that
AB
AD
=
AC
AE .
DB
EC
AD = AE .
AB
AC
AD = AE . Now,
Then, use algebra to show that
∆ADE ∼ ∆ABC which means that corresponding sides are proportional. Therefore,
you can use
algebra to show that the second proportion must be true. Remember that AB = AD + DB and AC = AE + EC.
AB
AD
AD + DB
→
AD
DB
→ 1+
AD
DB
→
AD
AC
AE
AE + EC
=
AE
EC
= 1+
AE
EC
=
AE
=
You have now proved the triangle proportionality theorem: a line parallel to one side of a triangle divides the
other two sides of the triangle proportionally.
Consider ∆ABC with AE the angle bisector of 6 BAC and point D constructed so that DCkAE. Prove that
2
EB
BA
=
EC
CA .
www.ck12.org
By the triangle proportionality theorem,
Chapter 1. Theorems Involving Similarity
EB
EC
=
BA
AD .
EC
BA
Multiply both sides of this proportion by
EC
BA .
EC
BA
EB
=
·
·
EC AD
BA
EB EC
→
=
BA AD
Now all you need to show is that AD = CA in order to prove the desired result.
•
•
•
•
∼ 6 EAC.
Because AE is the angle bisector of 6 BAC, 6 BAE =
∼
6
6
Because DCkAE, BAE = BDC (corresponding angles).
Because DCkAE, 6 EAC ∼
= 6 DCA (alternate interior angles).
∼
6
6
Thus, BDC = DCA by the transitive property.
Therefore, ∆ADC is isosceles because its base angles are congruent and it must be true that AD ∼
= CA. This means
that AD = CA. Therefore:
EB
BA
=
EC
CA
This proves the triangle angle bisector theorem: the angle bisector of one angle of a triangle divides the opposite
side of the triangle into segments proportional to the lengths of the other two sides of the triangle.
Examples
Example 1
Earlier, you were asked can you find any similar triangles in the picture below.
There are three triangles in this picture: ∆BAC, ∆BCD, ∆CAD. All three triangles are right triangles so they have one
set of congruent angles (the right angle). ∆BAC and ∆BCD share 6 B, so ∆BAC ∼ ∆BCD by AA ∼. Similarly, ∆BAC
and ∆CAD share 6 C, so ∆BAC ∼ ∆CAD by AA ∼. By the transitive property, all three triangles must be similar to
one another.
3
www.ck12.org
The large triangle above has sides a, b, and c. Side c has been divided into two parts: y and c − y. In the Concept
Problem Revisited you showed that the three triangles in this picture are similar.
Example 2
Explain why
a
c
=
c−y
a .
When triangles are similar, corresponding sides are proportional. Carefully match corresponding sides and you see
that ac = c−y
a .
Example 3
Explain why
b
c
= by .
When triangles are similar, corresponding sides are proportional. Carefully match corresponding sides and you see
that bc = by .
Example 4
Use the results from #2 and #3 to show that a2 + b2 = c2 .
Cross multiply to rewrite each equation. Then, add the two equations together.
a c−y
=
→ a2 = c2 − cy
c
a
b y
= → b2 = cy
c b
→ a2 + b2 = c2 − cy + cy
→ a2 + b2 = c2
You have just proved the Pythagorean Theorem using similar triangles.
Review
Solve for x in each problem.
1.
4
www.ck12.org
Chapter 1. Theorems Involving Similarity
2.
3.
4.
5
www.ck12.org
5.
6.
7.
Use the picture below for #8-#10.
6
www.ck12.org
Chapter 1. Theorems Involving Similarity
8. Solve for x.
9. Solve for z.
10. Solve for y.
Use the picture below for #11-#13.
11. Assume that
b
a
= dc . Use algebra to show that
b+a
a
=
d+c
c .
12. Prove that ∆Y ST ∼ ∆Y XZ
13. Prove that ST kXZ
14. Prove that a segment that connects the midpoints of two sides of a triangle will be parallel to the third side of the
triangle.
15. Prove the Pythagorean Theorem using the picture below.
Answers for Review Problems
To see the Review answers, open this PDF file and look for section 6.6.
7
www.ck12.org
References
1.
2.
3.
4.
5.
6.
7.
8
.
.
.
.
.
.
.
.
.
.
.
.
.
.
CC BY-NC-SA
CC BY-NC-SA
CC BY-NC-SA
CC BY-NC-SA
CC BY-NC-SA
CC BY-NC-SA
CC BY-NC-SA