1. Special Right Triangles
Section 8.3
Special Right Triangles

2. The diagonal of a square forms two congruent isosceles 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.

3. Example 1: Find the measure of each hypotenuse.
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.
45°-45°-90° Triangle Theorem
Substitution

4. 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.
45°-45°-90° Triangle Theorem
Substitution
x = 12

5. You can also work backwards using theorem 8
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.
45°-45°-90° Triangle Theorem
Substitution
Divide each side by
Rationalize the denominator.
Multiply.
Divide.

6. 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.
45°-45°-90° Triangle Theorem
Substitution
Divide each side by
Rationalize the denominator.
Multiply.
Divide.

7. A 30° – 60° – 90° triangle is another special right triangle
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.

8. 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.
30°-60°-90° Triangle Theorem
Substitution
Divide each side by 5.
30°-60°-90° Triangle Theorem
Substitution
Divide each side by 5.

9. 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.
30°-60°-90° Triangle Theorem
Substitution
Simplify.
30°-60°-90° Triangle Theorem
Substitution
Simplify.

10. 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.
30°-60°-90° Triangle Theorem
Substitution
Divide each side by
30°-60°-90° Triangle Theorem
Substitution
Simplify.

11. 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 length
of the side of one triangle. 3 ÷ 2 = 1.5
So the side length is 1.5 inches.
45°-45°-90° Triangle Theorem
Substitution

12. Example 6: Shaina designed 2 identical bookends according to the diagram below. Use special triangles to find the height of the bookends.
30°-60°-90° Triangle Theorem
Substitution