In these lessons, we will learn
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 these lessons, we will look at the 30º-60º-90º triangle. We also have lessons on other special right triangles
The 30º-60º-90º triangle is one example of a special right triangle. This is right triangle whose angles are 30º, 60ºand 90º. The lengths of the sides of a 30º-60º-90º triangle are in the ratio of
The hypotenuse is always twice the length of the shorter leg (the side facing the 30° angle). The longer leg (the side facing the 60° angle) is times of the shorter leg.
Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 4 inches and inches.
Step 1: Test the ratio of the lengths to see if it fits the ratio
Step 2: Yes, it is a 30-60-90 triangle with n = 4
Step 3: Calculate the third side.
2n = 2 × 4 = 8
Answer: The length of the hypotenuse is 8 inches.
You can also recognize a 30°-60°-90° triangle by the angles. As long as you know that one of the angles in the right-angle triangle is either 30° or 60° then it must be a 30°-60°-90° special right triangle.
A right triangle with a 30°-angle or 60°-angle must be a 30°-60°-90° special right triangle.
Find the lengths of the other two sides of a right triangle if the length
of the hypotenuse is 8 inches and one of the angles is 30°.
This is a right triangle with a 30°-60°-90° triangle.
You are given that the hypotenuse is 8. Substituting 8 into the third value of the ratio , we get that 2n = 8 Þ n = 4.
Substituting n = 4 into the first and second value of the ratio we get that the other two sides are
4 and .
Answer: The lengths of the two sides are 4 inches and inches.