Transcript 8-4 Similarity in Right Triangles

8-4
Similarity in Right Triangles
One Key Term
One Theorem
Two Corollaries
Theorem 8-3

Altitude Similarity Theorem
The altitude to the hypotenuse of a right
triangle divides the triangle into two
triangles that are similar to the original
triangle and to each other.
ABC ~ ACD ~ CBD
C
A
B
D
Vocabulary
1.
Geometric Mean
1.
a x

x b
x  ab
#1 Finding the Geometric Mean

Find the geometric mean of 15 and 20.
15 x

x 20
x  15(20)
x  300
x  10 3
Corollary 1 to Theorem 8-3
The length of the altitude to the
hypotenuse of a right triangle is the
geometric mean of the lengths of the
segments of the hypotenuse.
AD CD

CD DB
C
CD  AD(DB)
A
B
D
Corollary 2 to Theorem 8-3
The altitude to the hypotenuse of a right triangle
separates the hypotenuse so that the length of
each leg of the triangle is the geometric mean of
the length of the adjacent hypotenuse segment
and the length of the hypotenuse.
AD AC

,
AC AB
BD CB

CB AB
C
A
B
D
#2
• Solve for x and y.
Small Leg
4
Large Leg Hypotenusex
Small 
4
y
Medium 
y
12
Large 
x
x
12
y
16
4 x

x 16
4 y

y 12
x 2  64
y 2  48
x8
y4 3