Question

The length of the shorter leg of a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle is 24 meters. Find the length of the hypotenuse.

Short answer

The hypotenuse is 48 meters.

Step-by-step solution

Understanding the Problem

We are given a right triangle with angles \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\). The shorter leg, opposite the \(30^{\circ}\) angle, is 24 meters.

Identify the Relation in a 30-60-90 Triangle

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the relationship of the sides is as follows: the shorter leg (opposite the \(30^{\circ}\) angle) is \(x\), the longer leg (opposite the \(60^{\circ}\) angle) is \(x\sqrt{3}\), and the hypotenuse is \(2x\).

Substitute Known Values

Since the shorter leg \(x\) is 24 meters, we substitute \(x = 24\) into the hypotenuse formula, which is \(2x\).

Calculate the Length of the Hypotenuse

Using the formula for the hypotenuse \(2x\), substitute \(x = 24\) to get \(2 \times 24 = 48\).

Conclusion

Thus, the length of the hypotenuse is 48 meters.

Key concepts

Triangle Side Relationships

In geometry, triangle side relationships refer to how the lengths of a triangle's sides relate to each other. This is particularly important in solving problems that involve special triangles, like the 30-60-90 triangle. These relationships enable us to determine unknown side lengths by using known angles. The sides of a triangle are generally labeled according to their lengths relative to the angle they are opposite.
For a 30-60-90 triangle:

• The side opposite the 30-degree angle is considered the "shorter leg".
• The side opposite the 60-degree angle is the "longer leg".
• The side opposite the 90-degree angle, also known as the right angle, is the "hypotenuse".

The principle that the longest side (the hypotenuse in right triangles) is opposite the largest angle (90 degrees) is key to recognizing side relationships. Recognizing these side relationships allows us to apply specific rules or formulas to solve for missing sides.

Right Triangle

A right triangle is a type of triangle that has one of its angles measuring exactly 90 degrees. This specific angle is called the right angle. Right triangles are essential in mathematics and frequently appear in real-life applications. Because of the right angle, this type of triangle has unique characteristics that differentiate it from other triangles.
Key properties of a right triangle include:

• The side opposite the right angle is the longest side, known as the hypotenuse.
• The two sides that form the right angle are referred to as the legs of the triangle.
• Right triangles allow for the use of Pythagorean theorem. This theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides: \(a^2 + b^2 = c^2\).

These properties make right triangles a cornerstone of trigonometry and geometry. They provide useful skills for determining the values of sides and angles in various geometrical shapes and problems.

Special Triangles

Special triangles refer to triangles that have specific, predictable side length ratios. The 30-60-90 triangle is a perfect example of a special triangle. It's called "special" because it has fixed relationships between the lengths of its sides. These predictable ratios make solving problems much easier.
In a 30-60-90 triangle:

• The ratio of the side lengths is \(1 : \sqrt{3} : 2\).
• This makes calculations straightforward: if you know one side, you can figure out the other two.
• The shorter leg is half the hypotenuse: if \(x\) is the shorter leg, the hypotenuse is \(2x\).
• The longer leg can be found using the formula \(x\sqrt{3}\), where \(x\) is the shorter leg.

These triangles simplify problems because you only need to remember these ratios to solve for missing values. Understanding special triangles not only helps with computations but also builds a deeper understanding of geometric relationships.