5-8

Applying Special Right Triangles

Warm Up

Lesson Presentation

Lesson Quiz

Holt Geometry

Warm Up For Exercises 1 and 2, find the value of

x. Give your answer in simplest radical form. 1.

2. Simplify each expression. 3. 4.

Objectives

Justify and apply properties of 45-45-90

triangles. Justify and apply properties of 30-

60- 90 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 45-45-90 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.

(No Transcript)

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.

Check It Out! Example 1a

Find the value of x. Give your answer in simplest

radical form.

Simplify.

x 20

Check It Out! 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 tables 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...? Tessas other dog is wearing a square

bandana with a side length of 42 cm. What would

you expect the circumference of the other dogs

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).

y 2x

Simplify.

Check It Out! 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

Check It Out! Example 3b

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

simplest radical form.

y 2(5)

Simplify.

y 10

Check It Out! Example 3c

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

simplest radical form.

Hypotenuse 2(shorter leg)

24 2x

Divide both sides by 2.

12 x

Substitute 12 for x.

Check It Out! 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.

Hypotenuse 2(shorter leg)

6 2x

3 x

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.

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.

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.

Hypotenuse 2(shorter leg)

y 2x

Simplify.

Each side is approximately 34.6 cm.

Lesson Quiz Part I

Find the values of the variables. Give your

answers in simplest radical form. 1. 2.

3. 4.

x 10 y 20

Lesson Quiz Part II

Find the perimeter and area of each figure. Give

your answers in simplest radical form. 5. a

square with diagonal length 20 cm 6. an

equilateral triangle with height 24 in.