Question

1. a square and a rectangle have the same perimeter. find the side lengths of each figure. x + 1 2x - 3 x - 1

Explanation:

Step1: Find perimeter of square

The perimeter of a square with side length $s$ is $P = 4s$. Here $s=x + 1$, so $P_{square}=4(x + 1)=4x+4$.

Step2: Find perimeter of rectangle

The perimeter of a rectangle with length $l$ and width $w$ is $P = 2(l + w)$. Here $l=2x - 3$ and $w=x - 1$, so $P_{rectangle}=2((2x - 3)+(x - 1))=2(2x-3+x - 1)=2(3x - 4)=6x-8$.

Step3: Set perimeters equal

Since the perimeters are equal, $4x + 4=6x-8$.

Step4: Solve for x

Subtract $4x$ from both sides: $4=6x-4x - 8$, which simplifies to $4 = 2x-8$. Then add 8 to both sides: $4 + 8=2x$, so $12 = 2x$. Divide both sides by 2: $x = 6$.

Step5: Find side - lengths

For the square, side length $s=x + 1=6 + 1=7$. For the rectangle, length $l=2x - 3=2\times6-3=12 - 3 = 9$ and width $w=x - 1=6 - 1=5$.

Answer:

Side - length of square is 7; length of rectangle is 9 and width of rectangle is 5.