In these lessons, we will learn
A 45-45-90 triangle is a special right triangle whose angles are 45°, 45° and 90°.
The lengths of the sides of a 45-45-90 triangle are in the ratio of
1:1:√2.
The following diagram shows a 45-45-90 triangle and the ratio of its sides. Scroll down the page for more examples and solutions using the 45-45-90 triangle.
A right triangle with two sides of equal lengths is a 45-45-90 triangle.
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 a 45-45-90 special right triangle.
A right triangle with a 45° angle must be a 45-45-90 special right triangle.
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√2 is 3 then the length of the third side is 3√2
Answer: The length of the hypotenuse is 3√2 inches.
Example 2:
Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is 4√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√2. If the third value of the ratio n:n:n√2 is 4√2 then the lengths of the other two sides must 4.
Answer: The lengths of the two sides are both 4 inches.
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?Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
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