Problem 16 - Intermediate Algebra

Question

The lengths of two sides of the right triangle \(A B C\) are given. Find the length of the missing side. See Example 1 . (GRAPH CANT COPY) \(a=5\) in. and \(b=12\) in.

Step-by-Step Solution

Answer

The missing side, or the hypotenuse, is 13 inches.

Step 1: Identify the Problem Type

We are given a right triangle with two sides known and need to find the missing side. This situation calls for the use of the Pythagorean theorem.

Step 2: Apply the Pythagorean Theorem

The Pythagorean theorem states that for a right triangle with sides of length \( a \), \( b \), and hypotenuse \( c \), the equation is \( a^2 + b^2 = c^2 \). Here, \( a = 5 \) inches and \( b = 12 \) inches.

Step 3: Plug in the Known Values

Insert the given side lengths into the equation: \( 5^2 + 12^2 = c^2 \). First, calculate \( 5^2 = 25 \) and \( 12^2 = 144 \).

Step 4: Calculate the Sum

Add the squares of the two legs: \( 25 + 144 = 169 \). This is equal to \( c^2 \).

Step 5: Solve for the Hypotenuse

Take the square root of both sides to find \( c \): \( c = \sqrt{169} = 13 \) inches.

Key Concepts

Understanding Right Triangles

In geometry, a right triangle is a type of triangle that has one angle equal to 90 degrees. This special angle makes right triangles unique because it allows us to use specific mathematical principles to solve for unknown triangle sides. In a right triangle, the two sides that form the right angle are called "legs," and the side opposite to the right angle is the longest side, known as the "hypotenuse."
The Pythagorean theorem is an essential tool when working with right triangles. It provides a relationship between the lengths of the sides and allows us to calculate the length of the hypotenuse when the lengths of the other two sides are known. This theorem is not only crucial for solving problems involving geometry but also holds significance in various practical applications, such as construction and navigation.

The Role of the Hypotenuse

The hypotenuse in a right triangle is the longest side and always lies opposite the right angle. It plays a crucial role because its length can be calculated using the Pythagorean theorem. In our original problem, the hypotenuse is not directly given but can be determined.
When using the Pythagorean theorem, the equation is written as \(a^2 + b^2 = c^2\), where \(c\) represents the length of the hypotenuse. Thus, if you know the lengths of the two legs, you can solve for \(c\) by taking the square root of the sum of the squares of the other two sides. This calculation allows you to easily find the length of the hypotenuse, which, in our problem, was calculated to be 13 inches.

Determining Triangle Sides

Identifying and calculating the lengths of triangle sides in a right triangle often involves utilizing the Pythagorean theorem. With known values, like the two given side lengths in the problem \(a = 5\) inches and \(b = 12\) inches, you can plug these numbers into the equation \(5^2 + 12^2 = c^2\).
Step by step, this becomes \(25 + 144 = 169\). Next, the task is to determine the length of the missing hypotenuse by applying the inverse operation—taking the square root. Therefore, \(c = \sqrt{169} = 13\) inches.

• This systematic approach not only confirms the theory but also ensures precision in solving practical problems.
• By understanding how to apply these concepts confidently, you strengthen your foundation in geometry and mathematics effectively.