Question

A city planner designs a park that is a quadrilateral with vertices at j(-3,1), k(1,3), l(5,-1), 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

Answer 1

By definition, a quadrilateral is a two-dimensional figure that is composed of four sides. Since you are given with the coordinates of the vertices, the first logical thing to do is to plot the points in a Cartesian plane as shown in the attached picture. Next, find the midpoints of each side of the quadrilateral as denoted by the red dots. Then, create a line connecting the opposite midpoints which represents the length of the paths. 

Our first solution is to find the midpoint of each sides. This is done by taking the average of each coordinate. The equation would be:

x,midpoint = (x₁+x₂)/2
y,midpoint = (y₁+y₂)/2

So, applying the equation:

midpoint jk = (-3+1/2, 1+3/2) = (-1,2)
midpoint kl = (1+5/2, 3+⁻1/2) = (3,1)
midpoint ml = (5+⁻1/2, ⁻1+⁻3/2) = (2,-2)
midpoint jm = (⁻3+⁻1/2, 1+⁻3/2) = (-2,-1)

Next, we find the lengths of the paths by using the distance formula:

d = √(x₂ - x₁)² + (y₂ - y₁)²

Distance between jk and ml = √(⁻1-2)² + (2 - ⁻2)² = 5
Distance between jm and kl = √(3-⁻2)² + (1-⁻1)² = √29
Total distance = 5 + √29 = 10.385 units

Since the scale is 1 unit = 10 meters, the actual total length of paths is equal to:

Actual Total Distance = 10.385*10
Actual Total Distance = 103.85 meters