Question

1. the equilateral triangle and the square have the same perimeter.

image of square 1.5x + 12
image of equilateral triangle 2.5x + 10
a. what is the value of x? ______
b. what is the perimeter of each figure? ______

Explanation:

Part (a)

Step1: Recall perimeter formulas

The perimeter of a square with side length \( s \) is \( 4s \), and the perimeter of an equilateral triangle with side length \( t \) is \( 3t \). Here, the side of the square is \( 1.5x + 12 \), so its perimeter is \( 4(1.5x + 12) \). The side of the equilateral triangle is \( 2.5x + 10 \), so its perimeter is \( 3(2.5x + 10) \). Since their perimeters are equal, we set up the equation:
\[ 4(1.5x + 12) = 3(2.5x + 10) \]

Step2: Expand both sides

Expand the left side: \( 4\times1.5x + 4\times12 = 6x + 48 \)
Expand the right side: \( 3\times2.5x + 3\times10 = 7.5x + 30 \)
Now the equation is:
\[ 6x + 48 = 7.5x + 30 \]

Step3: Solve for \( x \)

Subtract \( 6x \) from both sides:
\[ 48 = 1.5x + 30 \]
Subtract 30 from both sides:
\[ 18 = 1.5x \]
Divide both sides by \( 1.5 \):
\[ x = \frac{18}{1.5} = 12 \]

Step1: Calculate perimeter of square

We know the side of the square is \( 1.5x + 12 \), and \( x = 12 \). Substitute \( x = 12 \) into the side length:
\( 1.5\times12 + 12 = 18 + 12 = 30 \)
The perimeter of the square is \( 4\times30 = 120 \) (we could also use the perimeter formula \( 4(1.5x + 12) \) and substitute \( x = 12 \): \( 4(1.5\times12 + 12)=4(18 + 12)=4\times30 = 120 \))

Step2: Verify with triangle's perimeter

The side of the triangle is \( 2.5x + 10 \), substitute \( x = 12 \):
\( 2.5\times12 + 10 = 30 + 10 = 40 \)
The perimeter of the equilateral triangle is \( 3\times40 = 120 \), which matches the square's perimeter, so it's correct.

Answer:

\( x = 12 \)

Part (b)