Question

1. the square and equilateral triangle below have the same perimeter. find the value of x. then, find the perimeter of each shape. 2\frac{1}{8}x + 30 5\frac{1}{3}x equation: __ x: perimeter: __

Explanation:

Step1: Write perimeter - equations

The perimeter of a square with side length \(s\) is \(P_{square}=4s\), and the perimeter of an equilateral triangle with side length \(t\) is \(P_{triangle}=3t\). Given \(s = 2\frac{1}{8}x+30=\frac{17}{8}x + 30\) and \(t = 5\frac{1}{3}x=\frac{16}{3}x\), the equation based on equal perimeters is \(4(\frac{17}{8}x + 30)=3(\frac{16}{3}x)\).

Step2: Expand the equation

Expand the left - hand side: \(4\times\frac{17}{8}x+4\times30=\frac{17}{2}x + 120\), and the right - hand side is \(16x\). So the equation becomes \(\frac{17}{2}x+120 = 16x\).

Step3: Get all \(x\) terms on one side

Subtract \(\frac{17}{2}x\) from both sides: \(120=16x-\frac{17}{2}x\). Find a common denominator for the right - hand side. \(16x=\frac{32}{2}x\), so \(120=\frac{32}{2}x-\frac{17}{2}x=\frac{32x - 17x}{2}=\frac{15}{2}x\).

Step4: Solve for \(x\)

Multiply both sides by \(\frac{2}{15}\): \(x = 120\times\frac{2}{15}=16\).

Step5: Find the perimeter

Substitute \(x = 16\) into the perimeter formula of the triangle (we could also use the square's formula). \(P = 3\times\frac{16}{3}\times16=256\).

Answer:

Equation: \(4(2\frac{1}{8}x + 30)=3(5\frac{1}{3}x)\)
x: \(16\)
Perimeter: \(256\)