Geometry HELP

Question

A city planner designs a park that is a quadrilateral with vertices at J(−3, 1), K(1, 3), L(5, −1), and M(−1, −1). 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.

The total length of the paths is approximately
meters.

Answer 1

The park is a quadrilateral with vertices J, K, L, & M, containing 4 entrances at the midpoint of each pair of vertices. (i.e. between J & K, K & L, L & M, M & J). So, you can start off this problem by finding the midpoint, or park entrance, coordinates for all four entrances. 

 

Remember, the midpoint formula is as follows:     m =  (X1 + X2)/2 , (Y1 + Y2)/2  

(this results in the coordinate of the midpoint in the form x,y)

 

For example: The midpoint between vertices J(-3,1) & K(1,3) can be found using the midpoint formula:

m = (-3 + 1 )/2 , (1 + 3)/2

m =  (-1,2)

 

The park entrances existing at the midpoint of the vertices, then, are as follows:

Entrance J-K : (-1,2)

Entrance K-L : (3,1)

Entrance L-M : (2,-1)

Entrance M-J : (-2,0)

 

Now, assuming you have this nicely graphed, it should be straightforward how to measure the distance of the paths connecting the entrances opposite one another. All you have to do now is use the distance formula to calculate how many units each opposite entrance is from one another. 

 

Distance Formula:  √( X2-X1)² + (Y2-Y1)²

 

Your answer will be the path length between Entrances at (-1,2) & (2,-1) as well as the path length between entrances at (3,1) & (-2,0). You will have two answers. Just remember whatever your answer comes out to, it is in graphical units (1 tick mark equals 1 unit), however, in this problem, one unit actually represents 10 meters. So, pretending you end up with 5 units as your answer, it will really equal 50 meters.