Altitudes

Recall that an altitude is a segment drawn from a vertex that is perpendicular to the opposite of a triangle. Every triangle has three altitudes.

Altitudes

In a right triangle, two of these altitudes are the two legs of the triangle. The other one is drawn perpendicular to the hypotenuse.

CA

B

D CA

B

D CA

B

Altitudes:

AB

BC

BD

Altitudes

Notice that this third altitude creates three right triangles. Is there something special about those triangles?

D CA

B

Altitudes:

AB

BC

BD

7.3 Use Similar Right Triangles

Objectives:

1. To find the geometric mean of two numbers

2. To find missing lengths in similar right triangles involving the altitude to the hypotenuse

Right Triangle Similarity Theorem

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.

Example 1

Identify the similar triangles in the diagram.

Example 2

Find the value of x.

Geometric Mean

The geometric mean of two positive numbers a and b is the positive number x that satisfies

This is just the square root of their product!

b

x

x

a

abx 2 So

abx And

Example 3

Find the geometric mean of 12 and 27.

Example 4

Find the value of x.

x

2712

Example 5

The altitude to the hypotenuse divides the hypotenuse into two segments.

What is the relationship between the altitude and these two segments?

x

2712

Geometric Mean Theorem I

Geometric Mean (Altitude) Theorem

In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments.

The length of the altitude is the geometric mean of the lengths of the two segments.

x

ba

b

x

x

a

Geometric Mean Theorem I

Geometric Mean (Altitude) Theorem

In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments.

The length of the altitude is the geometric mean of the lengths of the two segments.

x

ba

b

x

x

a

a b

xx

Heartbeat

Example 6

Find the value of w. 8 =w+9w+9 18

144 = (w+9)2

12 = w+9

w=3

Example 7

Find the value of x.

x

123

Geometric Mean Theorem II

Geometric Mean (Leg) Theorem

The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

x

a

a

c

x

c

a

y

c

b y

b

b

c

Geometric Mean Theorem II

Geometric Mean (Leg) Theorem

The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

x

a

a

c

x

c

a

y

c

b y

b

b

c

Boomerang

c

aa

x

Geometric Mean Theorem II

Geometric Mean (Leg) Theorem

The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

x

a

a

c

x

c

a

y

c

b y

b

b

c

Boomerang

c

bb

y

Example 8

Find the value of b.

Example 9

Find the value of variable.

1. w = 2. k =