Question

1. the perimeters of the square and equilateral triangle below are the same. write and solve an equation to find the value of x. then find the side lengths of each shape. 7x - 8.1 10x - 13

Explanation:

Step1: Set up the perimeter - equality equation

The perimeter of a square with side length \(s_1 = 7x - 8.1\) is \(P_{square}=4s_1 = 4(7x - 8.1)\). The perimeter of an equilateral triangle with side length \(s_2=10x - 13\) is \(P_{triangle}=3s_2 = 3(10x - 13)\). Since \(P_{square}=P_{triangle}\), we have the equation \(4(7x - 8.1)=3(10x - 13)\).

Step2: Expand both sides of the equation

Using the distributive property \(a(b + c)=ab+ac\), we get \(28x-32.4 = 30x-39\).

Step3: Isolate the variable \(x\)

First, subtract \(28x\) from both sides: \(-32.4=30x - 28x-39\), which simplifies to \(-32.4 = 2x-39\). Then add 39 to both sides: \(-32.4 + 39=2x\), so \(6.6 = 2x\). Finally, divide both sides by 2: \(x=\frac{6.6}{2}=3.3\).

Step4: Find the side - lengths of each shape

For the square, \(s_1=7x - 8.1\). Substitute \(x = 3.3\) into it: \(s_1=7\times3.3-8.1=23.1 - 8.1 = 15\). For the equilateral triangle, \(s_2=10x - 13\). Substitute \(x = 3.3\) into it: \(s_2=10\times3.3-13=33 - 13 = 20\).

Answer:

\(x = 3.3\), side - length of the square is \(15\), side - length of the equilateral triangle is \(20\)