Question

A city planner designs a park that is a quadrilateral with vertices at J(−3, 1), K(3, 3), L(5, −1), and M(−1, −3). There is an entrance to the park at the midpoint of each side of the park. A straight path connects each entrance to the entrance on the opposite side. Assuming each unit of the coordinate plane represents 10 meters, what is the total length of the paths to the nearest meter? Round your answer to the nearest whole number.

Answer 1

Answer: To determine the length of the paths connecting the entrance to the opposite entrance, we first need to find the midpoint of each side of the park. Then, we will use the distance formula to calculate the distance between each pair of midpoints.

Let's begin by finding the midpoint of each side of the park.

JK: (-3 + 3, 1 + 3) / 2 = (0, 2)

KL: (3 + 5, 3 - 1) / 2 = (4, 2)

LM: (5 - 1, -1 - -3) / 2 = (2, -2)

MJ: (-1 + -3, -3 + 1) / 2 = (-2, -1)

Next, we'll use the distance formula to calculate the distance between each pair of midpoints.

JK: sqrt((4 - 0)^2 + (2 - 2)^2) = sqrt(16 + 0) = 4

KL: sqrt((2 - 4)^2 + (-2 - 2)^2) = sqrt(16 + 16) = 4 sqrt(2)

LM: sqrt((-2 - 2)^2 + (-1 + 2)^2) = sqrt(16 + 9) = sqrt(25) = 5

MJ: sqrt((-2 - 0)^2 + (-1 - 2)^2) = sqrt(4 + 9) = sqrt(13)

The total length of the paths is 4 + 4 sqrt(2) + 5 + sqrt(13), which when rounded to the nearest whole number is 14.

Explanation: