Question

I need to figure out if the rectangle is a parallelogram by the coordinates of the midpoints

Answer 1

Based on the coordinates of the midpoints of the diagonals, ABCD is a parallelogram

Here, the diagonals we have are AC and BD

So, we get the midpoint of each

for AC; (-5,3) and (-1,-4)

We have the midpoint formula as follows;

$(x,y)\text{ = }(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\text{ = (}\frac{-5-1}{2},\frac{3-4}{2})=(-3,-0.5)$

For BD, we have;

(-7,-1) and (1,0)

We have the midpoint as follows;

$(x,y)\text{ = (}\frac{-7+1}{2},\frac{-1+0}{2})\text{ = (-3,-0.5)}$

Mathematically, the diagonals bisect each other if the shape is a parallelogram

By bisecting, they meet at the same point and the coordinates of their midpoints are same

From what we calculated above, this is the case and thus, Based on the coordinates of the midpoints of the diagonals, ABCD is a parallelogram