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.
Related topics
Composite Figures – Area and Volume
A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]
Read More >>Special Right Triangles: Types, Formulas, with Solved Examples.
Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem? Right Angle Triangles A triangle with a ninety-degree […]
Read More >>Ways to Simplify Algebraic Expressions
Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]
Read More >>How to Solve Right Triangles?
In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles. Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of […]
Read More >>Other topics

Comments: