Problem 39 - Algebra 2 and Trigonometry

Question

The lengths of the legs of a right triangle are 8 centimeters and 12 centimeters. Express the length of the hypotenuse in simplest radical form.

Step-by-Step Solution

Answer

The hypotenuse is \(4\sqrt{13}\) cm.

Step 1: Identify Triangle Elements

We are dealing with a right triangle with legs of 8 cm and 12 cm. We need to find the hypotenuse.

Step 2: Apply Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (\( c \)) is equal to the sum of the squares of the lengths of the other two sides. \( c^2 = a^2 + b^2 \). Substitute the given leg lengths: \( c^2 = 8^2 + 12^2 \).

Step 3: Calculate Square of Legs

Calculate the squares of each leg: \( 8^2 = 64 \) and \( 12^2 = 144 \). Add these results: \( c^2 = 64 + 144 = 208 \).

Step 4: Solve for Hypotenuse

Take the square root of both sides to solve for \( c \): \( c = \sqrt{208} \). Simplify the square root: \( c = \sqrt{16 \times 13} \). This becomes \( c = \sqrt{16} \times \sqrt{13} = 4\sqrt{13} \).

Key Concepts

Understanding Right Triangles

When we talk about right triangles, we're referring to a specific type of triangle that has one 90-degree angle. This angle is known as the right angle. In any right triangle, the side opposite the right angle is called the hypotenuse, and it is the longest side of the triangle. The other two sides, which form the right angle, are known as the legs.
Understanding which side is which is crucial for applying mathematical rules correctly. The Pythagorean Theorem is a powerful tool for working with right triangles. It allows us to find the length of one side if we know the lengths of the other two. Always remember: the hypotenuse is never one of the legs and is always directly opposite the right angle.

Simplifying Radicals

Simplifying radicals is an essential skill when dealing with square roots. It involves breaking down the number inside the square root into its prime factors. This process helps us find exact values in more manageable forms. For example, the number 208 can be expressed in terms of its prime factors, which are 16 and 13.
By simplifying \(\sqrt{208}\), we can split it as \(\sqrt{16 \times 13}\). Since 16 is a perfect square (\(4^2\)), it can be extracted from the square root: \(4\sqrt{13}\). The process of simplification not only makes the result cleaner but also helps in better understanding various mathematical operations.

Calculating the Hypotenuse

The hypotenuse calculation relies heavily on the Pythagorean Theorem in right triangles. This theorem states that the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides. For our problem, you use the given leg lengths (8 cm and 12 cm) to perform the calculation: \(c^2 = 8^2 + 12^2\).
Once the squares are calculated and added (64 and 144 resulting in 208), we take the square root to find the hypotenuse: \(c = \sqrt{208}\). To express this in simplest radical form, further simplify \(\sqrt{208}\) to \(4\sqrt{13}\), providing an exact and clear understanding of the hypotenuse's length. Knowing how to apply these steps will make working with right triangles straightforward.