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# Right Triangles

## Key Concepts

• Find hypotenuse length in a 45°-45°-90° triangle

### Special Right Triangles

There are two special right triangles with angles measures as 45°, 45°, 90° degrees and 30°, 60°, 90° degrees. The sides of these triangles are in particular ratios and are known as Pythagorean triplets.

### 45°-45°-90° Triangle

In a 45°-45°-90° triangle, both legs are congruent, and the length of the hypotenuse is √2

times the length of a leg.

### Find hypotenuse length in a 45°-45°-90° triangle

#### Hypotenuse

A hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle.

Example 1:

Find the length of the hypotenuse.

Solution:

By the Triangle Sum Theorem, the measure of the third angle must be 45°.

Then the triangle is a 45°-45°-90° triangle, so by theorem, the hypotenuse is

√2 times as long as each leg.

Hypotenuse = leg ×√2 45°-45°-90° Triangle Theorem

=  7√2 (Substitute)

Hypotenuse = leg√𝟐

Example 2:

Find the length of the hypotenuse.

Solution:

By the 45° -45°-90° triangle theorem, the length of the  hypotenuse is the length of a leg times √2

hypotenuse = leg × √2

= 3 × √2

The length of the Hypotenuse is 3√2

Example 3:

Find leg lengths in a 45°-45°-90° triangle.

Solution:

By the Base Angles Theorem and the Corollary to the Triangle Sum Theorem, the triangle is a 45°-45°-90° triangle.

hypotenuse = leg × √2 ( 45°-45°-90° Triangle Theorem  )

2√2  = x √2 (Substitute)

2√2/√2 = x√2/√2

2 = x    (Simplify)

x = 2

Example 4:

Find the length of a leg of 45° -45°- 90° triangle with a hypotenuse of length 10.

Solution:

hypotenuse = leg ×√2

10 = √x  ×2

x = 10/√2

x = 10/√2 * √2/√2

x = 10*√2/2

x = 5 √2

## Exercise

• Identify the legs and hypotenuse of the triangle.
• The square tile shown has painted corners in the shape of congruent 45°-45°-90° triangles. What is the value of x? What is the side length of the tile?
• Copy and complete the table.
• Is it possible to build a triangle using the given side lengths?

4, 4, and 7

• Find the length of a leg of 45° -45°- 90° triangle with a hypotenuse of length 6.
• Find the length of the hypotenuse in the 45° – 45° – 90° triangle. Write your answer in radical form.
• Find the length of a leg of a 45°-45°-90° triangle with a hypotenuse of length 22.
• Find the length of the hypotenuse.

### What have we learned

• Understand special right triangles.
• Understand 45°- 45° -90°  Triangle theorem
• Understand how to find the length of the hypotenuse.
• Understand  how to find the lengths of the legs in the triangle.

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