Transcript Lesson 7.4 Similarity in Right Triangles

7.4 Similarity in Right Triangles
In this lesson we will learn the relationship between different
parts of a right triangle that has an altitude drawn in it
Geometric Mean
Before we look at right triangles we will examine something
called the GEOMETRIC MEAN
Geometric Mean: The number x such that
are positive numbers
a x

x b
, where a, b, and x
If we solve we get x2=ab, so x  ab
Ex. Find the geometric mean between 9 and 4.
You could solve the proportion
9 x

x 4
x2=36
x=6
OR take the short cut
x  94
x  36
x=6
Geometric Mean
Geometric Mean: The number x such that
are positive numbers
a x

x b
, where a, b, and x
If we solve we get x2=ab, so x  ab
Ex. Find the geometric mean between 10 and 15.
You could solve the proportion
10
x

x
15
x2=150
x5 6
OR take the short cut
x  10  15
x  150
x5 6
Practice Problems
Geometric Mean: The number x such that
are positive numbers
a x

x b
, where a, b, and x
If we solve we get x2=ab, so x  ab
Put these two problems on your direction sheet
1. Find the geometric mean between 5 and 20
2. Find the geometric mean between 12 and 15.
Similarity in Right Triangles
Theorem 7-3: The altitude to the hypotenuse of a right triangle
divides the triangles into two triangles that are similar to the original
triangle and to each other.
Geometric Mean with Altitude
Corollary 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
6.75in
5.2 in
8.75in
So, since 6.75 is the altitude, it is the geometric mean of 5.2
and 8.75
6.75  5.2  8.75
5.2
6.75

6.75 8.75
Similarity in Right Triangles
Ex. Find the values of x in the following right triangles.
9
7
x is the geometric mean
of 9 and 7
x  97
x
x  63
x3 7
x
5
5 is the geometric mean
of x and 3
3
5  3x
25  3x
25 / 3  x
Practice Problems
Put these three problems on your direction sheet. Find y in
each picture.
4.
3.
y
2
8
5.
9
19
y
Geometric Mean
Second Corollary to Theorem 7-3: The altitude to the hypotenuse
separates the hypotenuse so that the length of each leg of the
triangle is the geometric mean of the lengths of the hypotenuse and
the length of the segment of the hypotenuse adjacent to the leg.
6
6
3
6 is the geometric mean of 3
and 12
3 is the part of the
hypotenuse closest to side
of 6. 12 is the whole
hypotenuse
6  3  12
3 6

6 12
Geometric Mean
f is the geometric mean of
10 and 12
Example.
10
f
2
f  10  12
f  120
f  2 30
w is the geometric mean of
2 and 9
w
w  29
2
7
w  18
w3 2
Practice Problems
Put these two problems on your direction sheet
A
7. Find w, j
8. Find w, j
C
w
8
12
A
4
D
5
B
C
D
j
w
B
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