Problem 22 - Elementary and Intermediate Algebra

Question

The lengths of two sides of the right triangle \(A B C\) are given. Find the length of the missing side. \(a=14 \mathrm{in.}\) and \(c=50 \mathrm{in.}\) (RIGHT TRIANGLE CAN'T COPY)

Step-by-Step Solution

Answer

The missing side of the triangle is 48 inches.

Step 1: Identify the Right Triangle Components

In a right triangle, the three sides consist of the two legs and the hypotenuse. The hypotenuse is the longest side, opposite the right angle. In this problem, side \(a\) is one of the legs with a length of 14 inches, and side \(c\) is the hypotenuse with a length of 50 inches.

Step 2: Apply the Pythagorean Theorem

The Pythagorean Theorem states that \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse. We need to find the length of side \(b\), another leg of the triangle. Rearrange the formula to solve for \(b\): \(b^2 = c^2 - a^2\).

Step 3: Substitute the Known Values

Substitute the known lengths of \(a\) and \(c\) into the equation: \(b^2 = 50^2 - 14^2\). Evaluate \(50^2 = 2500\) and \(14^2 = 196\), then subtract: \(b^2 = 2500 - 196\).

Step 4: Calculate \(b^2\)

Perform the subtraction to find \(b^2\): \(b^2 = 2304\).

Step 5: Solve for \(b\)

Find \(b\) by taking the square root of \(b^2\): \(b = \sqrt{2304}\). Calculate \(\sqrt{2304} = 48\).

Step 6: Conclude the Length of the Missing Side

The missing side, \(b\), of the triangle has a length of 48 inches.

Key Concepts

Understanding Right Triangles

A right triangle is a specific type of triangle that includes a 90-degree angle. This angle makes it possible to easily differentiate between the sides of the triangle.
Here’s how it works:

• The side opposite the right angle is the longest and is specifically called the hypotenuse.
• The remaining two sides are known as legs. They meet at the right angle.

A good way to visualize a right triangle is by thinking of a typical ramp or a set of stairs. The flat ground and the upward stretch of the ramp form the two legs, while the slope itself is the hypotenuse.
In the exercise given, we are told that side \(a\) is one of the legs measuring 14 inches, and side \(c\) is the hypotenuse with a length of 50 inches.
Right triangles have special properties, which make them easy to work with, especially using the Pythagorean Theorem.

The Role of the Hypotenuse

The hypotenuse is the crowning glory of the right triangle. Being the longest side, its length is crucial in calculations and some geometric principles.
When dealing with problems about right triangles, the hypotenuse is a key factor.
From the Pythagorean Theorem perspective:

• In the formula \(a^2 + b^2 = c^2\), the hypotenuse is always represented by \(c\).
• It is this side that unravels the lengths of the other two sides when at least one side is known.

In our problem, side \(c\) is given as 50 inches.
This crucial piece of information lets us apply the Pythagorean Theorem and find the missing side \(b\).
Remember: Knowing the hypotenuse can often mean quickly discovering the triangle's other components.

Steps to Solving Equations for Right Triangles

Solving equations is like detective work in math, where you piece together information to uncover unknowns.
In the context of right triangles, solving a problem using the Pythagorean Theorem involves a few straightforward steps:

• Identify what you know: In our case, we know one leg, \(a = 14\), and the hypotenuse, \(c = 50\).
• Apply the Pythagorean Theorem: The theorem is \(a^2 + b^2 = c^2\). Since we are finding \(b\), rearrange it to \(b^2 = c^2 - a^2\).
• Substitute and solve: Plug the known values into the equation \(b^2 = 2500 - 196\).
• Calculate: Do the mathematics to find \(b^2 = 2304\), then solve for \(b\) by taking a square root, giving \(b = 48\).

This structured approach helps make sense of the problem and lead to the solution efficiently. It is all about using what is given to unearth what is not yet known.