Question

1. an equilateral triangle and a rectangle have the same perimeter. find the side lengths of the equilateral triangle and the rectangle.

Explanation:

Step1: Calculate the perimeter of the equilateral triangle

The side - length of the equilateral triangle is \(x + 4\). The perimeter of an equilateral triangle \(P_{triangle}=3(x + 4)=3x+12\).

Step2: Calculate the perimeter of the rectangle

The length of the rectangle is \(x + 3\) and the width is \(x + 1\). The perimeter of a rectangle \(P_{rectangle}=2[(x + 3)+(x + 1)]=2(2x + 4)=4x+8\).

Step3: Set the perimeters equal

Since \(P_{triangle}=P_{rectangle}\), we have the equation \(3x + 12=4x+8\).

Step4: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(12=x + 8\). Then subtract 8 from both sides to get \(x = 4\).

Step5: Find the side - length of the equilateral triangle

Substitute \(x = 4\) into the side - length expression of the equilateral triangle. The side - length \(s=x + 4\), so \(s=4 + 4=8\).

Answer:

8