Question

equilateral triangle
square
writing a multistep equation
the perimeters of the square and the equilateral
triangle are equal.
which equation represents this scenario?
what is the value of x?
what is the perimeter of the polygons?
5x + 0.5
3x + 3

Explanation:

Step1: Set up the equation

Since the perimeters of the equilateral triangle and the square are equal. The perimeter of an equilateral triangle with side length \(a\) is \(P_{triangle}=3a\), and the perimeter of a square with side length \(b\) is \(P_{square} = 4b\). Here \(a = 5x+0.5\) and \(b=3x + 3\). So the equation is \(3(5x + 0.5)=4(3x + 3)\).

Step2: Expand both sides

Using the distributive property \(a(b + c)=ab+ac\).
For the left - hand side: \(3(5x+0.5)=3\times5x+3\times0.5 = 15x+1.5\).
For the right - hand side: \(4(3x + 3)=4\times3x+4\times3=12x + 12\).
The equation becomes \(15x+1.5=12x + 12\).

Step3: Solve for \(x\)

Subtract \(12x\) from both sides: \(15x-12x+1.5=12x-12x + 12\), which simplifies to \(3x+1.5=12\).
Subtract \(1.5\) from both sides: \(3x+1.5 - 1.5=12-1.5\), so \(3x=10.5\).
Divide both sides by \(3\): \(x=\frac{10.5}{3}=3.5\).

Step4: Find the perimeter

Substitute \(x = 3.5\) into the formula for the perimeter of the square (or the triangle).
Using the square's formula \(P = 4(3x + 3)\), substitute \(x = 3.5\): \(P=4(3\times3.5+3)=4(10.5 + 3)=4\times13.5 = 54\).

Answer:

• The equation representing the scenario is \(3(5x + 0.5)=4(3x + 3)\).
• The value of \(x\) is \(3.5\).
• The perimeter of the polygons is \(54\).