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Geometry 2H
KEY
Similarity Part I REVIEW
G-SRT.4. Learning Target: I can prove the following
theorems in narrative paragraphs, flow diagrams, in two
column format, and/or using diagrams without words: a
line parallel to one side of a triangle divides the other two
proportionally, and conversely; the Pythagorean Theorem
using triangle similarity.
2. Given the triangle below, prove that a line parallel to
one side of a triangle divides the other two
proportionally.
1. Given the triangle below, prove the Pythagorean
Theorem using similar triangles.
Prove:
Given: PY MK
𝑀𝑃
𝑃𝑊
=
𝐾𝑌
𝑌𝑊
B
D
Statement
A
C
1)
Statement
Reflexive Property
2) ∠𝐵𝐶𝐴 ≅ ∠𝐶𝐷𝐵 ≅
∠𝐶𝐷𝐴
3) ∆𝐴𝐵𝐶~∆𝐴𝐶𝐷;
∆𝐴𝐵𝐶~∆𝐶𝐵𝐷
All right angles are
congruent
𝑥
𝑎
=
𝑎
𝑐
5) 𝑐𝑥 = 𝑎2
6)
𝑏
𝑐
=
𝑐−𝑥
𝑏
1) Given
PY MK
Reasons
1) ∠𝐵 ≅ ∠𝐵; ∠𝐴 ≅ ∠𝐴
4)
Reasons
AA Similarity Theorem
2) ∠1 ≅ ∠3, ∠2 ≅ ∠4
2) Corresponding Angles
Postulate
3) ∆𝑊𝑀𝐾~∆𝑊𝑃𝑌
3) AA Similarity Theorem
4)
Definition of Similarity
5)
𝑀𝑊
𝑃𝑊
=
𝐾𝑊
4) Definition of Similarity
𝑌𝑊
𝑀𝑊 = 𝑀𝑃 + 𝑃𝑊
𝐾𝑊 = 𝐾𝑌 + 𝑌𝑊
5) Segment Addition
Postulate
𝑀𝑃+𝑃𝑊
6) Substitution Property
of Equality
Cross Product Property
Definition of Similarity
7) 𝑏 2 = 𝑐 2 − 𝑐𝑥
Cross Product Property
8) 𝑏 2 = 𝑐 2 − 𝑎2
Substitution Property
9) 𝑎2 + 𝑏 2 = 𝑐 2
Addition Property of
Equality
6)
7)
𝑃𝑊
𝑀𝑃
𝑃𝑊
=
=
𝐾𝑌+𝑌𝑊
𝐾𝑌
𝑌𝑊
𝑌𝑊
7) Subtraction Property of
Equality
Geometry 2H: Triangle Similarity REVIEW
G-SRT.5. Learning Target: I can solve problems using
similarity criteria for triangles. I can prove relationships
in geometry figures using similarity criteria for triangles.
Name: ______________________________________
5. Use the picture below to answer the following
questions.
3. Jose wants to find the height of a building. Jose is
standing 15 feet away from the tree. The tree is 12 feet
tall. The tree is 12 feet away from the building.
(a) Draw a picture with the information given.
(b) How are the two triangles similar? __AA ~______
(b) What is the height of the building? (Round to the
nearest foot.)
12 15

 15 x  324  x  21.6
x 27
answer : x  22 ft
4. Karen wanted to measure the height of her school's
flagpole. She placed a mirror on the ground 46 feet from
the flagpole, and then walked backwards until she was able
to see the top of the pole in the mirror. Her eyes were 5 feet
above the ground and she was 13 feet from the mirror.
Using similar triangles, find the height of the flagpole to
the nearest tenth of a foot. (Figures may not be drawn to
scale)
(a) Is there enough information to prove the two triangles
are similar? yes, by AA Similarity Theorem. A  A
by reflexive, ADE  ABC by corresponding angles.
(b) If so, find the value of x. If not, what additional
information would be needed?
x 13

 25 x  225  x  9
45 25
6. Given the two triangles shown below,
(a) What similarity method makes it possible to find the
value of x? Corresponding sides are proportional
5 13

 13x  230  x  17.7
x 46
answer : x  17.7 ft
(b) Find the value of x.
8
x

 12 x  240  x  20
12 30
Geometry 2H: Triangle Similarity REVIEW
G-GPE.6. Learning Target: I can find the point on a
directed line segment between two given points that
partitions the segment in a given ratio.
7. Line segment AB in the coordinate plane has endpoints
with coordinates A (−9,5) and 𝐵(1,0) Graph AB and
find the locations of point P so that P divides AB into
two parts with lengths in a ratio of 1:4.
NOTE: There are TWO possible answers. You must find
both for full credit.
Show all of your work.
A
P
P B
Name: ______________________________________
 ax  bx2 ay1  by2 
P 1
,

ab 
 ab
 1(9)  4(1) 1(5)  4(0) 
P
,

1 4 
 1 4
 9  4 5  0 
 
,

5 
 5
 5 5 
  , 
 5 5
P
 1,1
 1(1)  4(9) 1(0)  4(5) 
P
,

1 4 
 1 4
 1  (36) 0  20 
 
,

5
5 

 35 20 
 
, 
 5 4 
P   7, 4 