Questions: Consider the following word problem: Consider a square and a regular hexagon (a six-sided figure with sides of equal length). One side of the square is 19 feet longer than a side of the hexagon, and the two figures have the same perimeter. What are the lengths of the sides of each figure? Length of one side of the square: feet Length of one side of the hexagon: feet

Solution Steps

To solve this problem, we need to set up equations based on the given conditions. Let \( s \) be the side length of the hexagon. The side length of the square is then \( s + 19 \). The perimeter of the hexagon is \( 6s \) and the perimeter of the square is \( 4(s + 19) \). Since the perimeters are equal, we can set up the equation \( 6s = 4(s + 19) \) and solve for \( s \). Once we find \( s \), we can easily find the side length of the square.

Let \( s \) be the side length of the hexagon. The side length of the square is \( s + 19 \). The perimeter of the hexagon is \( 6s \) and the perimeter of the square is \( 4(s + 19) \). Since the perimeters are equal, we set up the equation: \[ 6s = 4(s + 19) \]

Simplify and solve the equation: \[ 6s = 4s + 76 \] Subtract \( 4s \) from both sides: \[ 2s = 76 \] Divide both sides by 2: \[ s = 38 \]

The side length of the square is \( s + 19 \): \[ s + 19 = 38 + 19 = 57 \]

Final Answer