Problem 49 - Algebra 2 and Trigonometry

Question

Find the length of the shorter leg of a right triangle if the length of the longer leg is 36 inches and the length of the hypotenuse is 39 inches.

Step-by-Step Solution

Answer

The length of the shorter leg is 15 inches.

Step 1: Identify the Formula

In a right triangle, the Pythagorean Theorem applies. This theorem states that \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the lengths of the two legs. In this problem, we know \(c = 39\) inches and the longer leg \(b = 36\) inches. We need to find \(a\).

Step 2: Substitute Known Values

Substitute the known values into the Pythagorean Theorem: \(a^2 + 36^2 = 39^2\). This requires calculating the squares of 36 and 39.

Step 3: Simplify the Equation

Calculate the squares: \(36^2 = 1296\) and \(39^2 = 1521\). The equation becomes \(a^2 + 1296 = 1521\).

Step 4: Solve for the Unknown Leg

Subtract 1296 from both sides of the equation to isolate \(a^2\): \(a^2 = 1521 - 1296\). Simplifying the right side gives \(a^2 = 225\).

Step 5: Calculate the Length of the Shorter Leg

Take the square root of both sides of the equation to find \(a\): \(a = \sqrt{225}\). This gives \(a = 15\).

Key Concepts

Understanding Right Triangles

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This specific angle makes it special and allows us to use unique mathematical rules. In a right triangle, the side opposite the right angle is called the hypotenuse. It is the longest side of the triangle. The other two sides are known as the legs. Each of these legs forms part of the right angle. Right triangles are very important in geometry and trigonometry because they help us understand properties and relationships between different parts of a triangle.

Some key points about right triangles include:

• The right angle is always present, making it a prominent feature.
• The hypotenuse is always longer than either leg.
• We can apply the Pythagorean Theorem to solve problems involving lengths of sides.

The Hypotenuse of a Right Triangle

The hypotenuse is a crucial side in a right triangle. As the longest side, it holds a special role in calculations involving the triangle. It stretches across from the right angle and connects the two legs. In practical terms, when looking at a triangle, the hypotenuse is always opposite the 90-degree angle.

To further understand:

• In the Pythagorean Theorem, the hypotenuse is denoted by the letter \(c\).
• It represents the diagonal distance directly across from the right angle.
• In our example, the hypotenuse is 39 inches long.

This length helps us determine the unknown leg of the triangle using mathematical principles.

Exploring the Legs of a Triangle

The legs of a right triangle are significant because they make up the right angle. These two sides work together to create that perfect 90-degree angle inside the triangle. Among these two legs, one can be longer than the other, or they might even be the same length.

• In most problems, we refer to these legs as \(a\) and \(b\).
• When using the Pythagorean Theorem, these legs are squared and summed up \(a^2 + b^2\).
• If one of the legs is unknown, like in our example, we can use known lengths to find the missing measure.

Understanding the role of each leg is critical to solving right triangle problems correctly.

Calculating with Square Roots

The process of finding a square root allows us to "work backward" from the square of a number. In terms of right triangles, square root calculations are essential in finding the length of a side when we have its square. The conclusion of the Pythagorean Theorem often involves taking a square root to solve for an unknown side.Here's how it works in practice:

• First, we calculate the squares of known lengths.
• Subsequently, we rearrange to isolate the squared unknown side.
• Finally, we take the square root to find the actual length.

In this case, for the length of the shorter leg, we calculated \(\sqrt{225}\), which simplified to 15. Thus, understanding square roots is not only helpful but essential in dealing with geometric problems involving right triangles.