5 8 Applying Special Right Triangles Objectives Justify

5 -8 Applying Special Right Triangles Objectives Justify and apply properties of 45°-90° triangles. Justify and apply properties of 30°- 60°- 90° triangles. Holt Geometry

5 -8 Applying Special Right Triangles A diagonal of a square divides it into two congruent isosceles right triangles. Since the base angles of an isosceles triangle are congruent, the measure of each acute angle is 45°. So another name for an isosceles right triangle is a 45°-90° triangle. A 45°-90° triangle is one type of special right triangle. You can use the Pythagorean Theorem to find a relationship among the side lengths of a 45°-90° triangle. Holt Geometry

5 -8 Applying Special Right Triangles Holt Geometry

5 -8 Applying Special Right Triangles Example 1 A: Finding Side Lengths in a 45°- 45º- 90º Triangle Find the value of x. Give your answer in simplest radical form. By the Triangle Sum Theorem, the measure of the third angle in the triangle is 45°. So it is a 45°-45° 90° triangle with a leg length of 8. Holt Geometry

5 -8 Applying Special Right Triangles Example 1 B: Finding Side Lengths in a 45º- 90º Triangle Find the value of x. Give your answer in simplest radical form. The triangle is an isosceles right triangle, which is a 45°-90° triangle. The length of the hypotenuse is 5. Rationalize the denominator. Holt Geometry

5 -8 Applying Special Right Triangles Check It Out! Example 1 b Find the value of x. Give your answer in simplest radical form. The triangle is an isosceles right triangle, which is a 45°-90° triangle. The length of the hypotenuse is 16. Rationalize the denominator. Holt Geometry

5 -8 Applying Special Right Triangles Check It Out! Example 1 a Find the value of x. Give your answer in simplest radical form. By the Triangle Sum Theorem, the measure of the third angle in the triangle is 45°. So it is a 45°-45° 90° triangle with a leg length of x = 20 Holt Geometry Simplify.

5 -8 Applying Special Right Triangles A 30°-60°-90° triangle is another special right triangle. You can use an equilateral triangle to find a relationship between its side lengths. Holt Geometry

5 -8 Applying Special Right Triangles Example 3 A: Finding Side Lengths in a 30º-60º-90º Triangle Find the values of x and y. Give your answers in simplest radical form. 22 = 2 x Hypotenuse = 2(shorter leg) 11 = x Divide both sides by 2. Substitute 11 for x. Holt Geometry

5 -8 Applying Special Right Triangles Check It Out! Example 3 c Find the values of x and y. Give your answers in simplest radical form. 24 = 2 x Hypotenuse = 2(shorter leg) 12 = x Divide both sides by 2. Substitute 12 for x. Holt Geometry

5 -8 Applying Special Right Triangles Check It Out! Example 3 b Find the values of x and y. Give your answers in simplest radical form. y = 2(5) y = 10 Holt Geometry Simplify.

5 -8 Applying Special Right Triangles Check It Out! Example 3 a Find the values of x and y. Give your answers in simplest radical form. Hypotenuse = 2(shorter leg) Divide both sides by 2. y = 27 Holt Geometry Substitute for x.

5 -8 Applying Special Right Triangles Example 3 B: Finding Side Lengths in a 30º-60º-90º Triangle Find the values of x and y. Give your answers in simplest radical form. Rationalize the denominator. y = 2 x Hypotenuse = 2(shorter leg). Simplify. Holt Geometry

5 -8 Applying Special Right Triangles Check It Out! Example 3 d Find the values of x and y. Give your answers in simplest radical form. Rationalize the denominator. x = 2 y Hypotenuse = 2(shorter leg) Simplify. Holt Geometry

5 -8 Applying Special Right Triangles Example 4: Using the 30º-60º-90º Triangle Theorem An ornamental pin is in the shape of an equilateral triangle. The length of each side is 6 centimeters. Josh will attach the fastener to the back along AB. Will the fastener fit if it is 4 centimeters long? Step 1 The equilateral triangle is divided into two 30°-60°-90° triangles. The height of the triangle is the length of the longer leg. Holt Geometry

5 -8 Applying Special Right Triangles Example 4 Continued Step 2 Find the length x of the shorter leg. 6 = 2 x 3=x Hypotenuse = 2(shorter leg) Divide both sides by 2. Step 3 Find the length h of the longer leg. The pin is approximately 5. 2 centimeters high. So the fastener will fit. Holt Geometry

5 -8 Applying Special Right Triangles Check It Out! Example 4 What if…? A manufacturer wants to make a larger clock with a height of 30 centimeters. What is the length of each side of the frame? Round to the nearest tenth. Step 1 The equilateral triangle is divided into two 30º-60º-90º triangles. The height of the triangle is the length of the longer leg. Holt Geometry

5 -8 Applying Special Right Triangles Check It Out! Example 4 Continued Step 2 Find the length x of the shorter leg. Rationalize the denominator. Step 3 Find the length y of the longer leg. y = 2 x Hypotenuse = 2(shorter leg) Simplify. Each side is approximately 34. 6 cm. Holt Geometry

5 -8 Applying Special Right Triangles Lesson Quiz: Part I Find the values of the variables. Give your answers in simplest radical form. 1. 2. x = 10; y = 20 3. Holt Geometry 4.