Question

A city planner designs a park that is a quadrilateral with vertices at J(−1, 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: The four vertices of the quadrilateral represent the corners of the park, and each side of the park has a midpoint that represents an entrance. To find the total length of the paths, we need to find the distance between each pair of opposite entrances.

Let's start by finding the coordinates of the midpoints of each side of the park.

Midpoint of JK: (J + K)/2 = ((−1, 1) + (3, 3))/2 = (1, 2)

Midpoint of KL: (K + L)/2 = ((3, 3) + (5, −1))/2 = (4, 1)

Midpoint of LM: (L + M)/2 = ((5, −1) + (−1, −3))/2 = (2, −2)

Midpoint of MJ: (M + J)/2 = ((−1, −3) + (−1, 1))/2 = (−1, −1)

Next, we can use the Pythagorean theorem to find the distance between each pair of opposite entrances.

Distance between entrance 1 and 3: sqrt((4 - 1)^2 + (1 + 2)^2) = sqrt(9 + 9) = 3√2

Distance between entrance 2 and 4: sqrt((−1 + 2)^2 + (−1 + 1)^2) = sqrt(9 + 4) = √13

Finally, we multiply each distance by 10 (since each unit of the coordinate plane represents 10 meters) and round the result to the nearest whole number.

Distance between entrance 1 and 3: 3√2 * 10 = 17 (rounded to nearest whole number)

Distance between entrance 2 and 4: √13 * 10 = 12 (rounded to nearest whole number)

The total length of the paths is the sum of these two distances: 17 + 12 = 29 (rounded to nearest whole number).

Therefore, the total length of the paths is 29 meters.

Explanation: