9.3 Altitude-On-Hypotenuse Theorems

Objective:After studying this section, you will be able to identify the relationships between the parts of a right triangle

when an altitude is drawn to the hypotenuse

ABC ACD CBD A B

C

D

When altitude CD is drawn to the hypotenuse of triangle ABC, three similar triangles are formed.

AB

C

D

by AA, notice thatABC ACD 2, or (AC) ( )( )

AB ACAB AD

AC AD

Therefore, AC is the mean proportional between AB and AD

AB

C

D

by AA, notice thatABC CBD 2, or (CB) ( )( )

AB CBAB DB

CB DB

Therefore, CB is the mean proportional between AB and DB

by transitivity of

similar triangles, notice that

ACD CBD

2, or (CD) ( )( )AD CD

AD DBCD DB

Therefore, CD is the mean proportional between AD and DB

CC

A D D B

Theorem If an altitude is drawn to the hypotenuse of a right triangle, then

a. The two triangles formed are similar to the given right triangle and to each other.

b. The altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse.

2, or x h

h xyh y

h

A B

C

Dy

b a

xc

c. Either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to that leg (i.e. the projection of that leg on the hypotenuse)

2, or y a

a cya c

h

AB

C

Dy

a b

xc

2, or x b

b cxb c

Example 1

If AD = 3 and DB = 9, find AC

A B

C

D

Example 2

If AD = 3 and DB = 9, find CD

If DB = 21 and AC = 10, find AD

A B

C

D

Example 3

Prove: (PO)(PM) = (PR)(PJ)J M

P

K

Given: O

R, ,PK JM RK JP KO PM

Summary:

Summarize what you learned from today’s lesson.

Homework:

Worksheet 9.3