45-45-90 Right Triangle



Recognizing special right triangles in geometry can help you to answer some questions quicker. A special right triangle is a right triangle whose sides are in a particular ratio. You can also use the Pythagorean theorem, but if you can see that it is a special triangle it can save you some calculations.

In this lesson, we will learn

  • the special right triangle called the 45°-45°-90° triangle.
  • how to solve problems involving the 45°-45°-90° right triangle

Related Topics: Other special right triangles

45º-45º-90º Triangles

A 45-45-90 triangle is a special right triangle whose angles are 45º, 45º and 90º. The lengths of the sides of a 45º-45º-90º triangle are in the ratio of 1:1:root 2 .

Note that a 45°-45°-90° triangle is an isosceles right triangle. It is also sometimes called a 45-45 right triangle.

A right triangle with two sides of equal lengths is a 45°-45°-90° triangle.

n:n:nroot2

 45-45-90 rt triangle

Solving problems with 45°- 45°- 90° triangles

Example 1:

Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are both 3 inches.

Solution:

Step 1: This is a right triangle with two equal sides so it must be a 45°-45°-90° triangle.

Step 2: You are given that the both the sides are 3. If the first and second value of the ratio n:n:n root 2  is 3 then the length of the third side is 3 root 2

Answer: The length of the hypotenuse is 3 root 2  inches.

 

You can also recognize a 45°-45°-90° triangle by the angles. As long as you know that one of the angles in the right-angle triangle is 45° then it must be a45°-45°-90° special right triangle.

A right triangle with a 45° angle must be a 45°-45°-90° special right triangle.

Example 2:

Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is 4 root 2 inches and one of the angles is 45°.

Solution:

Step 1: This is a right triangle with a 45° so it must be a 45°-45°-90° triangle.

Step 2: You are given that the hypotenuse is 4 root 2 . If the third value of the ratio n:n:n root 2  is 4 root 2 then the lengths of the other two sides must 4.

Answer: The lengths of the two sides are both 4 inches.  

Videos

The following videos show more examples of 45-45-90 triangles.

How to find the length of a leg or hypotenuse in a 45-45-90 triangle using the Pythagorean Theorem and then derive the ratio between the length of a leg and the hypotenuse.
This video gives an introduction to the 45-45-90 triangles and shows how to derive the ratio between the lengths of legs and the hypotenuse.



This video provides examples of how to solve a 45-45-90 triangle given the length of one side by using the ratio.






Special Right Triangles in Geometry: 45-45-90 and 30-60-90
This video discusses two special right triangles, how to derive the formulas to find the lengths of the sides of the triangles by knowing the length of one side, and then does a few examples using them.



Example problems of finding the sides of a 45-45-90 triangle with answer in simplest radical form.







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