Problem 216 - Elementary Algebra

Question

In the following exercises, solve using triangle properties. A triangular courtyard has perimeter 120 meters. The lengths of two sides are 30 meters and 50 meters. How long is the third side?

Step-by-Step Solution

Answer

The third side is 40 meters.

Step 1: Understand the given data

The problem provides the perimeter of the triangular courtyard as 120 meters, and the lengths of two sides as 30 meters and 50 meters.

Step 2: Recall the formula for the perimeter of a triangle

The perimeter of a triangle is calculated by adding up the lengths of all its sides. Mathematically, this is represented as: \[ P = a + b + c \] where \( P \) is the perimeter, and \( a, b, \) and \( c \) are the lengths of the sides.

Step 3: Substitute known values into the formula

From the problem, we know that \( P = 120 \) meters, \( a = 30 \) meters, and \( b = 50 \) meters. We need to find \( c \). Substituting the known values, we get: \[ 120 = 30 + 50 + c \]

Step 4: Solve for the unknown side

To isolate \( c \), subtract the sum of the known sides from the total perimeter: \[ c = 120 - (30 + 50) \] \[ c = 120 - 80 \] \[ c = 40 \]

Step 5: Write the final answer

The length of the third side is 40 meters.

Key Concepts

Perimeter of a Triangle

The perimeter of a triangle is the total distance around the triangle. To find this, we add up the lengths of all three sides. This simple formula can be very useful in many practical situations, such as finding the length of fencing needed to enclose a triangular garden. In mathematical terms, the formula is:

\[ P = a + b + c \]
where \( P \) stands for the perimeter, and \( a, b, \) and \( c \) are the lengths of the sides. In everyday problems, you'll often be given the perimeter and some of the side lengths. Knowing how to add and subtract using the perimeter formula makes it easy to find the unknown side.

For instance, if a triangular courtyard has a perimeter of 120 meters and two sides are 30 meters and 50 meters, you can simply follow the steps to calculate the third side.

Basic Algebra

Algebra involves using symbols and letters to represent numbers and quantities in formulas and equations. It's a foundation for more advanced math and is used extensively in most fields of science and engineering. In the problem at hand, once we have the perimeter formula set up, we use basic algebra to solve for the unknown side.

We start with the equation derived from the perimeter formula:

\[ 120 = 30 + 50 + c \]

Here, we simply substitute the known values from the problem. This is a straightforward substitution which is a key part of many algebraic solutions. Once the equation is set up with known values, we can move on to solving for the unknown.
At first glance, this might look complicated, but it's just basic addition and subtraction.

Solving for Unknown

Solving for the unknown means finding the value of an unknown variable. This is a common task in many algebra problems. Let's break it down using the example problem:

We started with the formula:
\[ 120 = 30 + 50 + c \]
We want to find \( c \), the length of the third side. To isolate \( c \), we subtract the sum of the known sides (30 and 50) from the perimeter. This looks like:
\[ c = 120 - (30 + 50) \]
We first calculate the sum inside the parentheses:
\[ c = 120 - 80 \]
Then we subtract 80 from 120:
\[ c = 40 \]
So, the length of the third side is 40 meters. By mastering these skills, you can tackle a wide range of problems with confidence.