CONFIDENTIAL1 Geometry Similarity in Right Triangles.

CONFIDENTIAL 1

Geometry

Similarity in Right Triangles

CONFIDENTIAL 2

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 =-4/32) x- intercept= -4 y-intercept =23) x- intercept= 5 y-intercept =-1

CONFIDENTIAL 3

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

DB

AC


CONFIDENTIAL 4

Theorem 1.2

DB

AC

Given: ∆ABC is a right triangle with altitude CD.Prove: ∆ABC ~ ∆ACD ~ ∆CBD

Proof: The right angles in ∆ABC, ∆ACD, and ∆CBDAre 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.

CONFIDENTIAL 5

Write a similarity statement comparing the three triangles.

Sketch the three right triangles with the angles of the triangles in corresponding positions.

SS

SR

R

T PP

P

T S

R

Identifying Similar Right Triangles

CONFIDENTIAL 6

Consider the proportion a = x. In this case, the means

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.

x b

CONFIDENTIAL 7

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


x2=(6)(15) = 90 x= 90 =3 10 Def. of geometric mean


CONFIDENTIAL 8


1a) 2 and 8

1b) 10 and 30

1c) 8 and 9

Now you try!

1a) 4 1b) 10√31c) 6√2

CONFIDENTIAL 9

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.

c

b =

b

h=

a

hc

a =

b

h =

a

x b

a =

y

h =

h

x

c

a

h

x

b

y

hh

y

x

b

a C

C

C

D

D D

B

B

AA

CONFIDENTIAL 10

Corollaries

COROLLARY EXAMPLE DIAGRAM

a

b

x

y

h

c

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.

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.

h2 = xy

a2 = xcb2 = yc

CONFIDENTIAL 11

Finding Side Lengths in right Triangles

Find x, y, and z.

x2 = (2) (10) = 20 x = 20 = 2 5 y2 = (12)(10) = 120

y = 120 = 2 30

z2 = (12)(2) = 24

z = 24 = 2 6

2

10

z

y

x

X is the geometric mean of 2 and 10.

Find the positive square root.

Y is the geometric mean of 12 and 10.


Z is the geometric mean of 12 and 2.


CONFIDENTIAL 12

2) Find u, v, and w.

9

3

vw

u

Now you try!

2) u = 27, v = 3√10, w = 9√10

CONFIDENTIAL 13

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?

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

Big Tex is about 47 + 5, or 52 ft tall.

Convert 3 in. to 0.25 ft. 15.25 is the geometric mean of 5 and x.

Solve for x and round.

5 ft

15 ft 3 in.

CONFIDENTIAL 14

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?

28 ft

Now you try!

3) 148 ft

5.5 ft

CONFIDENTIAL 15

Now some problems for you to practice !

CONFIDENTIAL 16

Assessment

Write a similarity statement comparing the three

triangles in each diagram.

SQR

PB

C

E D

1) 2)

CONFIDENTIAL 17

Find the geometric mean of each pair of numbers. If

necessary, give the answer in simplest radical form.

3). 2 and 50

4). 9 and 12

5). ½ and 8

3) 10 4) 6√35) 2

CONFIDENTIAL 18

Find x, y, and z.

zy

46

x

20

y

x

z 10

7).6).

6) x = 2√15, y = 2√6, z = 2√107) x = 5, y = 10√5, z = 5√5

CONFIDENTIAL 19

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?

4 m

60 m

S

T

U

R

8) 16√15

CONFIDENTIAL 20

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

DB

AC


Let’s review

CONFIDENTIAL 21

Theorem 1.2

DB

AC

Given: ∆ABC is a right triangle with altitude CD.Prove: ∆ABC ~ ∆ACD ~ ∆CBD

Proof: The right angles in ∆ABC, ∆ACD, and ∆CBDAre 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.

CONFIDENTIAL 22

Write a similarity statement comparing the three triangles.

Sketch the three right triangles with the angles of the triangles in corresponding positions.

SS

SR

R

T PP

P

T S

R

Identifying Similar Right Triangles

CONFIDENTIAL 23

Finding Geometric Means


A ) 4 and 9


x2 = (4)(9) = 36 x=6

Def. of geometric mean


B ) 6 and 15


x2=(6)(15) = 90 x= 90 =3 10 Def. of geometric mean


CONFIDENTIAL 24

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.

c

b =

b

h=

a

hc

a =

b

h =

a

x b

a =

y

h =

h

x

c

a

h

x

b

y

hh

y

x

b

a C

C

C

D

D D

B

B

AA

CONFIDENTIAL 25

Corollaries

COROLLARY EXAMPLE DIAGRAM

a

b

x

y

h

c

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.

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.

h2 = xy

a2 = xcb2 = yc

CONFIDENTIAL 26

Finding Side Lengths in right Triangles

Find x, y, and z.

x2 = (2) (10) = 20 x = 20 = 2 5 y2 = (12)(10) = 120

y = 120 = 2 30

z2 = (12)(2) = 24

z = 24 = 2 6

2

10

z

y

x

X is the geometric mean of 2 and 10.


Y is the geometric mean of 12 and 10.


Z is the geometric mean of 12 and 2.


CONFIDENTIAL 27

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?

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

Big Tex is about 47 + 5, or 52 ft tall.

Convert 3 in. to 0.25 ft. 15.25 is the geometric mean of 5 and x.

Solve for x and round.

5 ft

15 ft 3 in.

CONFIDENTIAL 28

You did a great job today!You did a great job today!

CONFIDENTIAL1 Geometry Similarity in Right Triangles.

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