SECTION 8.3

SPECIAL RIGHT TRIANGLES

The diagonal of a square forms two congruent isosceles right triangles. Since the base angles are congruent, the measure of each acute angle is 90 2, or 45°. Such a triangle is also known as a 45° – 45° – 90° triangle.

Example 1: Find the measure of each hypotenuse.

a)

The given angles of this triangle are 45° and 90°. This makes the third angle 45°, since 180 – 45 – 90 = 45. Thus, the triangle is a 45°-45°-90° triangle.

Substitution

45°-45°-90° Triangle Theorem

Example 1: Find the measure of each hypotenuse.

b)

The legs of this right triangle have the same measure, x, so it is a 45°-45°-90° triangle. Use the 45°-45°-90° Triangle Theorem.

Substitution

45°-45°-90° Triangle Theorem

x = 12

You can also work backwards using theorem 8.8 to find the legs of a 45° – 45° – 90° triangle.

Example 2: Find the measure of each leg.

a) The length of the hypotenuse of a 45°-45°-90° triangle is times as long as a leg of the triangle.

Substitution

45°-45°-90° Triangle Theorem

2

2h

8 2a

Multiply.

Divide.

Rationalize the denominator.

Divide each side by 28

2a

8 2

2 2a

8 2

2a

4 2 a

Example 2: Find the measure of each leg.

b) The length of the hypotenuse of a 45°-45°-90° triangle is times as long as a leg of the triangle.

Substitution

45°-45°-90° Triangle Theorem

2

2h

6 2a

Multiply.

Divide.

Rationalize the denominator.

Divide each side by 26

2a

6 2

2 2a

6 2

2a

3 2 a

A 30° – 60° – 90° triangle is another special right triangle. You can use an equilateral triangle to find this relationship. When the altitude is drawn from any vertex of an equilateral triangle, two congruent 30° – 60° – 90° triangles are formed. In the figure shown, ∆ABC ∆CBD, so AD CD. If CD = x, then AC = 2x. This leads to the next theorem.

Example 3: Find x and y.

a) The acute angles of a right triangle are complementary, so the measure of the third angle is 90 – 30 or 60. This is a 30°-60°-90° triangle.

Substitution

30°-60°-90° Triangle Theorem2h s

10 2x

Divide each side by 5.5x

Substitution

30°-60°-90° Triangle Theorem2h s

10 2y

Divide each side by 5.5y

Example 3: Find x and y.

b)

The acute angles of a right triangle are complementary, so the measure of the third angle is 90 – 30 or 60. This is a 30°-60°-90° triangle.

Substitution

30°-60°-90° Triangle Theorem3s

4 33

3x

Simplify.12or 4

3x

Substitution

30°-60°-90° Triangle Theorem2h s

4 32

3y

Simplify.8 3

3y

Example 4: Find the length of BC.

The acute angles of a right triangle are complementary, so the measure of the third angle is 90 – 30 or 60. This is a 30°-60°-90° triangle.

Substitution

30°-60°-90° Triangle Theorem

2 4BC

Divide each side by .4 AB

Substitution

30°-60°-90° Triangle Theorem2h s

Simplify.8BC

3s

4 3 3AB

3

Example 5: A quilt has the design shown in the figure, in which a square is divided into 8 isosceles right triangles. If the length of one side of the square is 3 inches, what are the dimensions of each triangle?

Divide the length of the side of the square by 2 to find the lengthof the side of one triangle. 3 ÷ 2 = 1.5So the side length is 1.5 inches.

45°-45°-90° Triangle Theorem

Substitution

Example 6: Shaina designed 2 identical bookends according to the diagram below. Use special triangles to find the height of the bookends.

Substitution

30°-60°-90° Triangle Theorem3s

5 3x