Solving for the Hypotenuse in a 30-60-90 Triangle with Given Shorter Leg

Introduction to the 30-60-90 Triangle

In geometry, a 30-60-90 triangle is a special right triangle where the angles measure exactly 30°, 60°, and 90°. The sides of a 30-60-90 triangle have a specific ratio, making it easier to find the lengths of the sides given one side. The sides are as follows:

The shorter leg opposite the 30° angle is x. The longer leg opposite the 60° angle is x√3. The hypotenuse opposite the 90° angle is 2x.

Problem: Finding the Hypotenuse

A common problem involves finding the length of the hypotenuse when the length of the shorter leg is known. For example, let's consider the following problem:

The shorter leg of a 30-60-90 triangle is 6. What is the length of the hypotenuse?

Given that the shorter leg (opposite the 30° angle) is 6, we can find the hypotenuse as follows:

The length of the hypotenuse is given by the ratio 2x, where x is the length of the shorter leg. Therefore, hypotenuse 2x 2 × 6 12.

So, the length of the hypotenuse is 12 units.

Exploring Trigonometric Ratios

Another approach involves using trigonometric ratios, specifically the sine function. The sine of 30° is equal to the length of the opposite side (the shorter leg) divided by the hypotenuse:

sin(30°) opp/hyp

Given sin(30°) 0.5, we can write:

0.5 12 / hypotenuse

Solving for the hypotenuse, we get:

hypotenuse 2 × 12 24 units.

Alternative Approach: Decomposing an Equilateral Triangle

If you are not familiar with the properties of a 30-60-90 triangle, you can still find the hypotenuse by decomposing an equilateral triangle. An equilateral triangle is a triangle with all sides equal and each angle measuring 60°. If we bisect an equilateral triangle, we decompose it into two 30-60-90 right triangles.

Let's consider an equilateral triangle with a side length of 2:

Bisect one of the angles (60°) into 30° and 30°. This line also bisects the opposite side, creating a right angle with the base. The bisected side is split into two equal segments of length 1.

In the resulting 30-60-90 right triangle:

The shorter leg (opposite the 30° angle) is 1. The longer leg (opposite the 60° angle) is 1√3. The hypotenuse is the original side length of the equilateral triangle, which is 2.

From these observations, we can conclude that the hypotenuse of a 30-60-90 triangle is twice the length of the shorter leg. If the shorter leg is 12 cm, the hypotenuse is 24 cm.

Conclusion

In summary, understanding the properties of a 30-60-90 triangle and using trigonometric ratios can help solve problems involving the hypotenuse. Whether you use the side ratios or trigonometric methods, the length of the hypotenuse in a 30-60-90 triangle is consistently twice the length of the shorter leg.