9514 1404 393
Answer:
Y = (-1, 5)
Explanation:
The diagonals of a parallelogram bisect each other, so have the same midpoint. That is ...
(M +R)/2 = (A +Y)/2
Y = M + R - A . . . . . . multiply by 2 and subtract A
Y = (-3, 1) +(5, 7) -(3, 3) = (-3 +5 -3, 1 +7, -3) . . . . . substitute point coordinates
Y = (-1, 5)
Coordinate geometry was used to find Y that makes MARY a parallelogram. The fact that there is a solution is proof enough that MARY is a parallelogram.
_____
Your teacher may expect the "proof" to be either or both of ...
- showing opposite sides have the same slope
- showing opposite line segments have the same length
Given the method used above, both of these are unnecessary. However, in the interest of completeness, we will demonstrate.
Slope and Length of MA
slope = (3 -1)/(3 -(-3)) = 2/6 = 1/3
length = √(2²+6²) = √40
Slope and Length of AR
slope = (7 -3)/(5 -3) = 4/2 = 2
length = √(4² +2²) = √20
Slope and Length of RY
slope = (5 -7)/(-1 -5) = -2/-6 = 1/3 . . . . same as opposite side MA
length = √((-2)²+(-6)²) = √40 . . . . . . . same as opposite side MA
Slope and Length of YM
slope = (1 -5)/(-3 -(-1)) = -4/-2 = 2 . . . . same as opposite side AR
length = √((-4)² +(-2)²) = √20 . . . . . . same as opposite side AR
Opposite sides have the same slopes, so the figure is a parallelogram.
Opposite sides have the same lengths, so the figure is a parallelogram.