Transcript Slide 1

Geometry
Similarity in Right
Triangles
CONFIDENTIAL
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Warm up
Find the x- intercept and y-intercept for each equation.
1). 3y + 4 = 6x
2). x + 4 = 2y
3). 3y – 15 = 15x
1) x- intercept= 2/3 y-intercept =-4/3
2) x- intercept= -4 y-intercept =2
3) x- intercept= 5 y-intercept =-1
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Similarity in Right Triangles
In a right triangle, an altitude drawn from the vertex of the
right angle to the hypotenuse forms two right triangles.
Theorem 1.1
The altitude to the hypotenuse of a right triangle
forms two triangles that are similar to each other
and to the original triangle.
∆ABC ~ ∆ACD ~ ∆CBD
B
D
A
C
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Theorem 1.2
Given: ∆ABC is a right triangle with altitude CD.
Prove: ∆ABC ~ ∆ACD ~ ∆CBD
Proof: The right angles in ∆ABC, ∆ACD, and ∆CBD
Are all congruent.
By the Reflexive Property of
 
Congruence, A ≅ A. Therefore ∆ABC~ ∆ACD
 by
the AA Similarity Theorem. Similarly, B ≅ B, so
∆ABC ~ ∆CBD. By the Transitive Property of
Similarity, ∆ABC~∆ACD~∆CBD.
B
D
A
C
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Identifying Similar Right Triangles
Write a similarity statement comparing the three triangles.
Sketch the three right triangles with the angles
of the triangles in corresponding positions.
R
P
T
S
R
S
S T
P
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R
S
P
5
Consider the proportion a = x. In this case, the means
x
b
of the proportion are the same number, and that number is
the geometric mean of the extremes. The geometric mean
of two positive numbers is the positive square root of their
product. So the geometric mean of a and b is the positive
number x such that x = √ab, or x2=ab.
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Finding Geometric Means
Find the geometric mean of each pair of numbers. If
necessary, give the answer in simplest radical form.
A ) 4 and 9
Let x be the geometric mean.
x2 = (4)(9) = 36
x=6
Def. of geometric mean
Find the positive square root
B ) 6 and 15
Let x be the geometric mean.
x2=(6)(15) = 90
x= 90 =3 10
Def. of geometric mean
Find the positive square root
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Now you try!
Find the geometric mean of each pair of numbers. If
necessary, give the answer in simplest radical form.
1a) 2 and 8
1b) 10 and 30
1c) 8 and 9
1a) 4
1b) 10√3
1c) 6√2
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Theorem 1.1: to write proportions comparing the
side lengths of the triangles formed by the altitude to
the hypotenuse of a right triangle. All the
relationships in red involve geometric means.
B
x
D
C
c

y
a
h
C
b
a
A
b
=
y
h
=

b
h
D
h
c
x
a
A
y
=
b
h
=
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B
a
x
D
a
c
x
b
C
h
=
b
h
=
a
h
9
Corollaries
COROLLARY
EXAMPLE
1.2 The length of the
altitude to the hypotenuse
of a right triangle is the
geometric mean of the
lengths of two segments of
the hypotenuse.
DIAGRAM
h2 = xy
x
c
y
a
1.3 The length of a leg of a
right triangle is the
geometric mean of the
lengths of the hypotenuse
and the segment of the
hypotenuse adjacent to
that leg.
h
a2 = xc
b2 = yc
CONFIDENTIAL
b
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Finding Side Lengths in right Triangles
Find x, y, and z.
z
x
x2 = (2) (10) = 20
x=
20 = 2 5
y2 = (12)(10) = 120
y = 120 = 2 30
z = (12)(2) = 24
2
2
X is the geometric mean of 2 and 10.
Find the positive square root.
10
y
Y is the geometric mean of 12 and 10.
Find the positive square root.
Z is the geometric mean of 12 and 2.
Find the positive square root.
z = 24 = 2 6
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Now you try!
2) Find u, v, and w.
u
w
3
9
v
2) u = 27, v = 3√10, w = 9√10
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Measurement
Application
To estimate the height of Big Tex at the
State Fair of Texas, Michael steps away
from the statue until his line of sight to the
top of the status and his line of sight to
the bottom of the statue form a 90˚ angle.
His eyes are 5 ft above the ground, and he
is standing 15 ft 3 in. from Big Tex. How
tall is Big Tex to the nearest foot?
15 ft 3 in.
5 ft
Let x be the height of Big Tex above eye level.
15 ft 3 in. = 15.25 ft
(15.25)
= 5x
X = 46.5125 = 47
Convert 3 in. to 0.25 ft.
15.25 is the geometric mean
of 5 and x.
Solve for x and round.
Big Tex is about 47 + 5, or 52 ft tall.
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Now you try!
3) A surveyor positions himself so that his line of
sight to the top of a cliff and his line of sight to the
bottom from a right angle as shown. What is the
height of the cliff to the nearest foot?
3) 148 ft
28 ft
5.5 ft
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Now some problems for you to practice !
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Assessment
Write a similarity statement comparing the three
triangles in each diagram.
2)
1)
P
R
S
B
C
Q
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D
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Find the geometric mean of each pair of numbers. If
necessary, give the answer in simplest radical form.
3). 2 and 50
3) 10
4) 6√3
5) 2
4). 9 and 12
5). ½ and 8
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Find x, y, and z.
6).
7).
6
4
y
x
y
z
z
10
x
20
6) x = 2√15, y = 2√6, z = 2√10
7) x = 5, y = 10√5, z = 5√5
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8) Measurement To estimate the length of the US
Constitution in Boston harbor, a student located points
T and U as shown. What is RS to the nearest tenth?
R
S
60 m
U
T
4 m
8) 16√15
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Let’s review
Similarity in Right Triangles
In a right triangle, an altitude drawn from the vertex of the
right angle to the hypotenuse forms two right triangles.
Theorem 1.1
The altitude to the hypotenuse of a right triangle
forms two triangles that are similar to each other
and to the original triangle.
∆ABC ~ ∆ACD ~ ∆CBD
B
C
D
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A
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Theorem 1.2
Given: ∆ABC is a right triangle with altitude CD.
Prove: ∆ABC ~ ∆ACD ~ ∆CBD
Proof: The right angles in ∆ABC, ∆ACD, and ∆CBD
Are all congruent.
By the Reflexive Property of
 
Congruence, A ≅ A. Therefore ∆ABC~ ∆ACD
 by
the AA Similarity Theorem. Similarly, B ≅ B, so
∆ABC ~ ∆CBD. By the Transitive Property of
Similarity, ∆ABC~∆ACD~∆CBD.
B
D
A
C
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Identifying Similar Right Triangles
Write a similarity statement comparing the three triangles.
Sketch the three right triangles with the angles
of the triangles in corresponding positions.
R
P
T
S
R
S
S T
P
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R
S
P
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Finding Geometric Means
Find the geometric mean of each pair of numbers. If
necessary, give the answer in simplest radical form.
A ) 4 and 9
Let x be the geometric mean.
x2 = (4)(9) = 36
x=6
Def. of geometric mean
Find the positive square root
B ) 6 and 15
Let x be the geometric mean.
x2=(6)(15) = 90
x= 90 =3 10
Def. of geometric mean
Find the positive square root
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Theorem 1.1: to write proportions comparing the
side lengths of the triangles formed by the altitude to
the hypotenuse of a right triangle. All the
relationships in red involve geometric means.
B
x
D
C
c

y
a
h
C
b
a
A
b
=
y
h
=

b
h
D
h
c
x
a
A
y
=
b
h
=
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B
a
x
D
a
c
x
b
C
h
=
b
h
=
a
h
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Corollaries
COROLLARY
EXAMPLE
1.2 The length of the
altitude to the hypotenuse
of a right triangle is the
geometric mean of the
lengths of two segments of
the hypotenuse.
DIAGRAM
h2 = xy
x
c
y
a
1.3 The length of a leg of a
right triangle is the
geometric mean of the
lengths of the hypotenuse
and the segment of the
hypotenuse adjacent to
that leg.
h
a2 = xc
b2 = yc
CONFIDENTIAL
b
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Finding Side Lengths in right Triangles
Find x, y, and z.
z
x
x2 = (2) (10) = 20
x=
20 = 2 5
y2 = (12)(10) = 120
y = 120 = 2 30
z = (12)(2) = 24
2
2
X is the geometric mean of 2 and 10.
Find the positive square root.
10
y
Y is the geometric mean of 12 and 10.
Find the positive square root.
Z is the geometric mean of 12 and 2.
Find the positive square root.
z = 24 = 2 6
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Measurement
Application
To estimate the height of Big Tex at the
State Fair of Texas, Michael steps away
from the statue until his line of sight to the
top of the status and his line of sight to
the bottom of the statue form a 90˚ angle.
His eyes are 5 ft above the ground, and he
is standing 15 ft 3 in. from Big Tex. How
tall is Big Tex to the nearest foot?
15 ft 3 in.
5 ft
Let x be the height of Big Tex above eye level.
15 ft 3 in. = 15.25 ft
(15.25)
= 5x
X = 46.5125 = 47
Convert 3 in. to 0.25 ft.
15.25 is the geometric mean
of 5 and x.
Solve for x and round.
Big Tex is about 47 + 5, or 52 ft tall.
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You did a great job today!
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