Special Right Triangles

Objectives:

1. To use the properties of 45-45-90 and 30-60-90 right triangles to solve problems

Investigation 1

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.

Investigation 1

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

Investigation 1

Did you notice something interesting about the relationship between the length of the hypotenuse and the length of the legs in each problem of this investigation?

Special Right Triangle Theorem45°-45°-90° Triangle

Theorem

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

2

2leg hypotenuse

Example 1

Use deductive reasoning to verify the Isosceles Right Triangle Conjecture.

Example 2

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

Investigation 2

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 2

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?~

Investigation 2

Notice that altitude CD divides the equilateral triangle into two right triangles with acute angles that measure 30° and 60°. Look at just one of the 30-60-90 right triangles. How do AC and AD compare?

Conjecture:In a 30°-60°-90° right triangle, if the side

opposite the 30° angle has length x, then the hypotenuse has length -?-.

Investigation 2

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

Investigation 2

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

Investigation 2

You should have notice a pattern in your answers. Combine your observations with you latest conjecture and state your next conjecture.

Special Right Triangle Theorem30°-60°-90° Triangle

Theorem

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

Two Special Right Triangles

Example 3

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

1. 2. 3.