Special Right Triangles Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

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

Solution

  1. 1
    Step 1: Recall the 30-60-90 ratio: short leg : long leg : hypotenuse =1:3:2= 1 : \sqrt{3} : 2.
  2. 2
    Step 2: If the hypotenuse is 2k2k, then k=k = hypotenuse/2=16/2=8/2 = 16/2 = 8.
  3. 3
    Step 3: The short leg (opposite 30°) =k=8= k = 8.
  4. 4
    Step 4: The long leg (opposite 60°) =k3=83≈13.86= k\sqrt{3} = 8\sqrt{3} \approx 13.86.

Answer

Short leg =8= 8; Long leg =83≈13.86= 8\sqrt{3} \approx 13.86.
The 30-60-90 triangle has sides in ratio 1:3:21:\sqrt{3}:2. The shortest side is opposite the 30° angle and equals half the hypotenuse. The side opposite 60° is 3\sqrt{3} times the shortest side. These ratios come from an equilateral triangle cut in half by an altitude.

About Special Right Triangles

Two families of right triangles whose side ratios can be determined exactly: the 30-60-90 triangle with sides in ratio 1:3:21 : \sqrt{3} : 2, and the 45-45-90 triangle with sides in ratio 1:1:21 : 1 : \sqrt{2}.

Learn more about Special Right Triangles →

More Special Right Triangles Examples

Example 1 easy

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

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 rel

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