ABCD is a parallelogram where A(x, y), B(5, 8), C(4, 7) and D(2, -4). Find

1) Coordinates of A

2) An equation of diagonal BD

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#### Solution

1) We know that the diagonals of a parallelogram bisect each other

So, coordinates of mid-point of `BD = ((5+2)/2 , (8 -4)/2) = (7/2, 2) ` .....(1)

Now the mid-point of diagonal AC = `((x + 4)/2 , (y + 7)/2)` .....(2)

From (1) and (2), we get`((x + 4)/2 , (y + 7)/2) = (7/2, 2)`

Comparing, we get x + 4 = 7 and y + 7 = 4

Thus x = 3 and y = -3

So, the coordinates of point A are (3, -3)

2) Equation of a line is given by `y - y_1 = (y_2 - y_1)/(x_2 - x_1) (x - x_1)`

Coordinates of point B and D are (5, 8) and (2, -4) respectively.

Equation of a diagonal BD, y - 8 = `(-4-8)/(2-5) (x - 5)`

`=> y - 8 = 4(x - 5)`

or 4x - y = 12

Concept: The Mid-point of a Line Segment (Mid-point Formula)

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