7.4 similarity in right triangles in this lesson we will learn the relationship between different...
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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
Geometric Mean: The number x such that , where a, b, and x are positive numbers
If we solve we get x2=ab, so
a
xx
b
abx
Before we look at right triangles we will examine something called the GEOMETRIC MEAN
You could solve the proportion OR take the short cut
x2=36 x=6 x=6
4
9 x
x 49x
36x
Ex. Find the geometric mean between 9 and 4.
Geometric Mean
Geometric Mean: The number x such that , where a, b, and x are positive numbers
If we solve we get x2=ab, so
a
xx
b
abx
You could solve the proportion OR take the short cut
x2=150
15
10 x
x
65x
1510x150x
65x
Ex. Find the geometric mean between 10 and 15.
Practice Problems
Geometric Mean: The number x such that , where a, b, and x are positive numbers
If we solve we get x2=ab, so
a
xx
b
abx
Put these two problems on your direction sheet1.Find the geometric mean between 5 and 202.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
5.2 in 8.75in
6.75in
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
75.82.575.6
75.8
75.6
75.6
2.5
So, since 6.75 is the altitude, it is the geometric mean of 5.2 and 8.75
Similarity in Right TrianglesEx. Find the values of x in the following right triangles.
9 7
3
73x
63x79x
5
x
x
x325 x35
x3/25
x is the geometric mean of 9 and 7
5 is the geometric mean of x and 3
Practice ProblemsPut these three problems on your direction sheet. Find y in each picture.
3.
28
y4.
19
y
9
5.
Geometric MeanSecond 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
6 is the geometric mean of 3 and 123 is the part of the hypotenuse closest to side of 6. 12 is the whole hypotenuse
31236
12
6
6
3
Geometric Meanf is the geometric mean of 10 and 12
2f
10Example.
1210f
120f
302f
2 7
w
w is the geometric mean of 2 and 9
18w
92w
23w
Practice Problems
7. Find w, j
A
C
D B
w
54
jA
CDB
w
12
8
j
8. Find w, j
Put these two problems on your direction sheet
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