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Three of the vertices of quadrilateral MARY are M (-3,1), A(3,3), and R(5,7). Find the
coordinates of Y, such that, quadrilateral MARY is a parallelogram.

Use coordinate geometry to prove that MARY is a parallelogram.

Three Of The Vertices Of Quadrilateral MARY Are M 31 A33 And R57 Find The Coordinates Of Y Such That Quadrilateral MARY Is A Parallelogram Use Coordinate Geomet class=

Sagot :

Sqdancefanidn

9514 1404 393

Answer:

  Y = (-1, 5)

Step-by-step 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.

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