Problem 4 - Algebra for College Students

Question

A square has perimeter 160 in. What would be the perimeter of an equilateral triangle whose sides each measure the same length as the side of the square?

Step-by-Step Solution

Answer

The perimeter of the equilateral triangle is 120 inches.

Step 1: Determine the Side Length of the Square

First, recall the formula to calculate the perimeter of a square: \[ P = 4s \] where \(P\) is the perimeter and \(s\) is the side length. Given the perimeter \(P = 160 \) inches, we can solve for \(s\) using: \[ s = \frac{P}{4} = \frac{160}{4} = 40 \text{ inches} \]

Step 2: Confirm the Side Length

Ensure the side length of the square is correctly calculated as 40 inches by reevaluating with the perimeter formula: \[ P = 4s = 4 \times 40 = 160 \text{ inches} \]. This confirms our side length as 40 inches.

Step 3: Calculate the Perimeter of the Equilateral Triangle

The side length of the equilateral triangle is the same as the side length of the square, which is 40 inches. The perimeter of an equilateral triangle can be calculated using: \[ P = 3s \] where \(s\) is the side length. Thus: \[ P = 3 \times 40 = 120 \text{ inches} \]

Key Concepts

perimeter of a square

The perimeter of a square is the total distance around the square. To calculate it, you simply need to know the length of one of its sides and apply the formula: \( P = 4s \) where\(P\)stands for the perimeter and\(s\)represents one side length. For example, if a square has a perimeter of 160 inches, you can find the side length by dividing the perimeter by 4: \( s = \frac{P}{4} = \frac{160}{4} = 40 \) inches. This means each side of the square is 40 inches long. Always ensure your calculations are correct by using the formula again to verify.

perimeter of an equilateral triangle

An equilateral triangle is a triangle where all three sides are of equal length. The perimeter of an equilateral triangle is found by adding up the lengths of all its three sides. To make it easier, you can use the formula: \[ P = 3s \]where \(P\) stands for the perimeter and \(s\) represents the side length. For instance, if each side of the equilateral triangle measures 40 inches (the same as the side length of our square in the example), the perimeter of the triangle would be: \[ P = 3 \times 40 = 120 \text{ inches} \] This demonstrates that the triangle's perimeter is 120 inches when each side is 40 inches long.

side length

The side length is a crucial element in calculating the perimeter of geometric shapes. For a square, the side length is found by dividing the perimeter by 4. For example, if the perimeter is 160 inches, the side length would be \( s = \frac{160}{4} = 40 \text{ inches} \). Once you have the side length, it can be used to find the perimeter of other shapes like an equilateral triangle. To compute the perimeter of an equilateral triangle with the same side length, you multiply by 3: \[ P = 3s = 3 \times 40 = 120 \]. Remember, accurately determining the side length is essential for solving perimeter problems correctly.