9.1 Similar Right Triangles

Theorem

If an altitude is drawn to the hypotenuse of a Right triangle, then it makes similar triangles to the original Right triangle.

ACDCBD

ABCACD

ABCCBD

A

B

C

D

Since this is true

We have proportions.

ACDCBD

ABCACD

ABCCBD

A

B

C

D

BDABBC

BDABBCBC

AB

BD

BC

2

DBADAC

DBADACAC

DB

AD

AC

2

Since this is true

The legs of the right triangle are the Geometric mean to the part of the hypotenuse and the whole hypotenuse.

ACDCBD

ABCACD

ABCCBD

DBADAC

DBADACAC

DB

AD

AC

2

BDABBC

BDABBCBC

AB

BD

BC

2

Find x and y

Set up the proportions

12 8

x y

Find x and y

Set up the proportions

12 8

x yx

x 812

12

y

y 128

8

Find x and y

Set up the proportions

12 8

x y

154240

2012

812

12

2

x

x

x

x

104160

208

128

8

2

y

y

y

y

Since this is true

The altitude is also a Geometric mean.

ACDCBD

ABCACD

ABCCBD

DBADCD

DBADCDCD

DB

AD

CD

2

A

C

D

B

12

8

6496

812

CD

CD

Recapping

The Altitude is the geometric mean of the parts of the hypotenuse

A leg of the Original Right Triangle is the geometric mean of part of the hypotenuse touching it and the whole hypotenuse.

Find x, y and z

x yz

615

Find x, y and z

x yz

615

48.9103

90615

2.11143

1266156

7.17353

31561515

z

y

x

• Solve for x, y and z

20

18

x

y

z

20

18

18

20 x

• Solve for x, y and z

20

18

x

y

z

2.49

38

1876

32418400

181820

20

18

18

20

2

x

x

x

x

x

• Solve for x, y and z

20

18

y

z 2.4

y

y 2.418

2.4

• Solve for x, y and z

20

18

y

z 2.4

6.92.222.4

2.222.4

2.418

2.4

2

y

y

y

y

• Solve for x, y and z

20

18

y

z 2.4

7.82.418 z

Homework

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