Transcript 9.3 Altitude-On-Hypotenuse Theorems

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
When altitude CD is drawn to the hypotenuse
of triangle ABC, three similar triangles are
formed.
C
ABC
ACD
CBD
D
A
ABC ACD by AA, notice that
AB AC
2

, or (AC)  ( AB)( AD)
AC AD
B
C
A
D
B
Therefore, AC is the mean proportional between AB and AD
C
ABC CBD by AA, notice that
AB CB
2

, or (CB)  ( AB)( DB)
CB DB
A
D
B
Therefore, CB is the mean proportional between AB and DB
ACD CBD by transitivity of
similar triangles, notice that
C C
AD CD

, or (CD) 2  ( AD)( DB)
CD DB
A
DD
B
Therefore, CD is the mean proportional between AD and DB
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.
C
b
A
y
x h
2
 , or h  xy
h y
a
h
D
x
c
B
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)
y a
 , or a 2  cy
a c
C
a
B
y
b
h
D
x
c
A
x b
2
 , or b  cx
b c
Example 1
If AD = 3 and DB = 9, find CD
C
A
Example 2
If AD = 3 and DB = 9, find AC
D
B
Example 3
If DB = 21 and AC = 10, find AD
C
A
D
B
P
O
Given:
PK  JM , RK  JP, KO  PM
R
J
Prove: (PO)(PM) = (PR)(PJ)
K
M
Summary:
Summarize what you learned
from today’s lesson.
Homework:
Worksheet 9.3