Given the following 3 vertices, F(-5,1), A(-2,5), C(6,-1), find the fourth vertex, E, to make the figure a rectangle. Prove that FACE is a rectangle.

  E(3, -5)

Step-by-step explanation:

In a rectangle, the diagonals are the same length and bisect each other. That means their midpoints are the same. Then ...

  (F +C)/2 = (A +E)/2

  E = F +C -A

  E = (-5, 1) +(6, -1) -(-2, 5) = (-5+6+2, 1-1-5)

  E = (3, -5) . . . . . . . E is chosen so that the midpoint of AE is that of FC

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To prove the figure is a rectangle, we can show the lengths of the diagonals are the same. Using the distance formula, ...

  FC = √((6-(-5))^2 +(-1-1)^2) = √(11^2 +2^2) = √125

  AE = √((3-(-2))^2 +(-5-5)^2) = √(5^2 +10^2) = √125

The diagonals are the same length and have the same midpoint, so the figure is a rectangle.

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