Question

The square and equilateral triangle below have the same perimeter. Find the value of x. You must show ALL work to earn full credit.

(HINT: an equilateral triangle has 3 sides that are the same length)

Triangle:2x-2
Square: 2.5x-3

Answer 1

To find the value of \( x \) such that the perimeter of the square is equal to the perimeter of the equilateral triangle, we start by calculating the perimeters for each shape.

**Step 1: Calculate the perimeter of the equilateral triangle.**

The perimeter of an equilateral triangle is the sum of the lengths of its three equal sides. Given that each side of the triangle is \( 2x - 2 \), the perimeter \( P_{\text{triangle}} \) is:

\[
P_{\text{triangle}} = 3 \times (2x - 2)
\]

Now simplify this expression:

\[
P_{\text{triangle}} = 6x - 6
\]

**Step 2: Calculate the perimeter of the square.**

The perimeter of a square is the sum of the lengths of its four equal sides. Given that each side of the square is \( 2.5x - 3 \), the perimeter \( P_{\text{square}} \) is:

\[
P_{\text{square}} = 4 \times (2.5x - 3)
\]

Now simplify this expression:

\[
P_{\text{square}} = 10x - 12
\]

**Step 3: Set the perimeters equal to each other.**

Since the perimeters are equal, we set the two perimeter expressions equal to each other:

\[
6x - 6 = 10x - 12
\]

**Step 4: Solve for \( x \).**

To solve for \( x \), first, we can rearrange the equation:

\[
6x - 10x = -12 + 6
\]

This simplifies to:

\[
-4x = -6
\]

Now, divide both sides by -4:

\[
x = \frac{-6}{-4} = \frac{6}{4} = \frac{3}{2}
\]

**Step 5: Conclusion.**

Thus, the value of \( x \) is:

\[
\boxed{\frac{3}{2}}
\]