Geometry BGeometry BChapter 8Chapter 8

Lesson: Special Right Triangles

Warm UpFor Exercises 1 and 2, find the value of x. Give your answer in simplest radical form.

1. 2.

Simplify each expression.

3. 4.

Justify and apply properties of 45°-45°-90° triangles.

Justify and apply properties of 30°- 60°- 90° triangles.

Objectives

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°-45°-90° triangle.

A 45°-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°-45°-90° triangle.

Example 1A: 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.

Example 1B: Finding Side Lengths in a 45º- 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°-45°-90° triangle. The length of the hypotenuse is 5.

Rationalize the denominator.

In Your Notes! Example 1a

Find the value of x. Give your answer in simplest radical form.

x = 20 Simplify.

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

In Your Notes! Example 1b

Find the value of x. Give your answer in simplest radical form.

The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 16.

Rationalize the denominator.

Example 2: Craft Application

Jana is cutting a square of material for a tablecloth. The table’s diagonal is 36 inches. She wants the diagonal of the tablecloth to be an extra 10 inches so it will hang over the edges of the table. What size square should Jana cut to make the tablecloth? Round to the nearest inch.

Jana needs a 45°-45°-90° triangle with a hypotenuse of 36 + 10 = 46 inches.

Check It Out! Example 2

What if...? Tessa’s other dog is wearing a square bandana with a side length of 42 cm. What would you expect the circumference of the other dog’s neck to be? Round to the nearest centimeter.

Tessa needs a 45°-45°-90° triangle with a hypotenuse of 42 cm.

A 30°-60°-90° triangle is another special right triangle. You can use an equilateral triangle to find a relationship between its side lengths.

Example 3A: Finding Side Lengths in a 30º-60º-90º Triangle

Find the values of x and y. Give your answers in simplest radical form.

Hypotenuse = 2(shorter leg)22 = 2x

Divide both sides by 2.11 = x

Substitute 11 for x.

Example 3B: 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.

Hypotenuse = 2(shorter leg).

Simplify.

y = 2x

In Your Notes! Example 3a

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 Substitute for x.

In Your Notes! Example 3b

Find the values of x and y. Give your answers in simplest radical form.

Simplify.

y = 2(5)

y = 10

In Your Notes! Example 3c

Find the values of x and y. Give your answers in simplest radical form.

Hypotenuse = 2(shorter leg)

Divide both sides by 2.

Substitute 12 for x.

24 = 2x

12 = x

In Your Notes! Example 3d

Find the values of x and y. Give your answers in simplest radical form.

Rationalize the denominator.

Hypotenuse = 2(shorter leg)x = 2y

Simplify.

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.

Example 4 Continued

Step 2 Find the length x of the shorter leg.

Step 3 Find the length h of the longer leg.

The pin is approximately 5.2 centimeters high. So the fastener will fit.

Hypotenuse = 2(shorter leg)6 = 2x

3 = x Divide both sides by 2.

In Your Notes! 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.

In Your Notes! Example 4 Continued

Step 2 Find the length x of the shorter leg.

Each side is approximately 34.6 cm.

Step 3 Find the length y of the longer leg.

Rationalize the denominator.

Hypotenuse = 2(shorter leg)y = 2x

Simplify.