Special Right Triangles

Lesson 7-3

Investigation

This triangle is also referred to as a 45-45-90 right triangle because each of its acute angles measures 45°. Folding a square in half can make one of these triangles.

In this investigation, you will discover a relationship between the lengths of the legs and the hypotenuse of an isosceles right triangle.

Find the length of the hypotenuse of each isosceles right triangle. Simplify the square root each time to reveal a pattern.

Special Right Triangle Theorem

45°-45°-90° Triangle 45°-45°-90° Triangle TheoremTheorem

In a 45°-45°-90° triangle, the hypotenuse is times as long as each leg.

Verify….

2

2leg hypotenuse

Finding the Length of a Leg

Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 22.

Example

A fence around a square garden has a perimeter of 48 feet. Find the approximate length of the diagonal of this square garden.

Finding the Length of a Leg The distance from one corner to the

opposite corner of a square playground is 96 ft. To the nearest foot, how long is each side of the playground?

Investigation

The second special right triangle is the 30-60-90 right triangle, which is half of an equilateral triangle.

Let’s start by using a little deductive reasoning to reveal a useful relationship in 30-60-90 right triangles.

Investigation

Triangle ABC is equilateral, and segment CD is an altitude.

1. What are m<A and m<B?

2. What are m<ADC and m<BDC?

3. What are m<ACD and m<BCD?

4. Is ΔADC = ΔBDC? Why?

5. Is AD=BD? Why?

~

Find the length of the indicated side in each right triangle by using the conjecture you just made.

Now use the previous conjecture and the Pythagorean formula to find the length of each indicated side.

Special Right Triangle Theorem

30°-60°-90° Triangle 30°-60°-90° Triangle TheoremTheorem

In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.

3

legshorter 2 hypotenuse 3legshorter leglonger

Finding the Length of the Legs

Find the lengths of the legs of a 30°-60°-90° triangle with hypotenuse of length 8.

60

30

8

Finding the Length of the Legs Find the lengths of the legs of a 30°-

60°-90° triangle with hypotenuse of length 43.

60

30

43

Using the Length of a Leg The longer leg of a 30°-60°-90°

triangle has length 18. Find the lengths of the shorter leg and the hypotenuse.

Two Special* Right Triangles

*what’s so special about them?

ExampleFind the value of each variable. Write

your answer in simplest radical form.

1. 2. 3.

Example

Find the value of each variable. Write your answer in simplest radical form.

1. 2. 3.

Example

What is the area of an equilateral triangle with a side length of 4 cm?

4 cm

4 cm 4 cm

Finding Area

Challenge! Find all side lengths

Classwork

P. 3691-20, 21-29 odd, 34-40