Question

1. a square has side lengths of 2x inches. an equilateral triangle has side lengths of (2x + 1/3) inches. if the square and the triangle have the same perimeter, what is the value of x? (2x + 1/3) in. 2x in.

Explanation:

Step1: Calculate the perimeter of the square

The perimeter of a square is \(4\times\) side length. Given the side length of the square is \(2x\) inches, so the perimeter \(P_{square}=4\times(2x)=8x\) inches.

Step2: Calculate the perimeter of the equilateral triangle

The perimeter of an equilateral triangle is \(3\times\) side length. Given the side length of the equilateral triangle is \((2x +\frac{1}{3})\) inches, so the perimeter \(P_{triangle}=3\times(2x+\frac{1}{3})=6x + 1\) inches.

Step3: Set up the equation based on equal perimeters

Since \(P_{square}=P_{triangle}\), we have the equation \(8x=6x + 1\).
Subtract \(6x\) from both sides: \(8x-6x=6x + 1-6x\).
This simplifies to \(2x=1\).
Divide both sides by \(2\): \(x=\frac{1}{2}\).

Answer:

\(x = \frac{1}{2}\)