Practice Special Right Triangles in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

Two families of right triangles whose side ratios can be determined exactly: the 30-60-90 triangle with sides in ratio 1 : \sqrt{3} : 2, and the 45-45-90 triangle with sides in ratio 1 : 1 : \sqrt{2}.

Cut an equilateral triangle in half and you get a 30-60-90 triangle. Cut a square along its diagonal and you get a 45-45-90 triangle. These two cuts give you exact side ratios you can memorize forever—no calculator needed.

Example 1

easy
A 45-45-90 triangle has legs of length 7. Find the length of the hypotenuse.

Example 2

medium
In a 30-60-90 triangle, the hypotenuse is 16. Find the lengths of both legs.

Example 3

easy
The diagonal of a square is 10 cm. Find the side length of the square.

Example 4

hard
An equilateral triangle has side length 12. Find its height and area using the 30-60-90 triangle relationship.
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