Question

question 18 (4 points) the square and equilateral triangle below have the same perimeter. find the value of x. you must show all handwritten work to earn full credit. (hint: an equilateral triangle has 3 sides that are the same length)

Explanation:

Step1: Find the perimeter of the square

The side - length of the square is $2.5x - 3$. The perimeter of a square $P_{square}$ is given by $4$ times the side - length, so $P_{square}=4(2.5x - 3)=10x-12$.

Step2: Find the perimeter of the equilateral triangle

The side - length of the equilateral triangle is $2x - 2$. The perimeter of an equilateral triangle $P_{triangle}$ is given by $3$ times the side - length, so $P_{triangle}=3(2x - 2)=6x-6$.

Step3: Set the perimeters equal

Since the perimeters are equal, we set up the equation $10x-12 = 6x-6$.

Step4: Solve the equation for x

Subtract $6x$ from both sides: $10x-6x-12=6x - 6x-6$, which simplifies to $4x-12=-6$.
Add $12$ to both sides: $4x-12 + 12=-6 + 12$, so $4x = 6$.
Divide both sides by $4$: $x=\frac{6}{4}=\frac{3}{2}=1.5$.

Answer:

$x = 1.5$