Transcript Chapter 7: Similarity

Chapter 7: Similarity
7.4
Similarity in Right Triangles
Review
• Draw a right triangle. Label the triangle ABC,
with C as the right angle.
• Draw the altitude to the hypotenuse, and label it
CD.
• Name the two smaller right triangles that are
formed.
Theorem 7-3
• 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.
A
ABC ~ ACD ~ CBD
D
C
B
Geometric Mean
• the number x such that:
a x

x b
• where a, b, and x are positive numbers
• example:
• the geometric mean of 6 and 24 is 12:
6
x

x 24
x  144
2
Example 1
• Find the geometric mean of 4 and 18.
Example 1a
• Find the geometric mean of 15 and 20.
Corollary 1 to Theorem 7-3
• 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.
A
6
D
24
12
C
B
Corollary 2 to Theorem 7-3
• 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.
A
15
3
D
3 5
12
6
C
6 5
B
Example 2
• Solve for x and y.
x
y
4
5
Example 3
• Solve for x and y.
4
x
y
12
Example 4
• Solve for x:
x
40
50
Example 5
• Find x, y, and z.
z
x
y
1
4
Example 6
• Find x, y, and z.
z
x
y
5
4
Example 7
• Find the value of x:
x+2
5
x
Example 8
• Find the value of x:
x+3
x
12
Example 9
• Find the value of x:
2x + 1
x
8
Example 10
• Find the values of the variables:
6
30
z
y
x
Homework
• p. 394:
• 2-20 even, 26-36 even