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.

• 45°-45°-90° TRIANGLES: Find the value of x. Write your answer in the simplest radical form.

• 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 values of the variable(s). Write your answer(s) in the simplest radical form.

• 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.

Concept Map

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.