187k views
0 votes
Three vertices of a parallelogram are shown in the figure below.

Give the coordinates of the fourth vertex.

Three vertices of a parallelogram are shown in the figure below. Give the coordinates-example-1
User VsfDawg
by
3.8k points

1 Answer

1 vote

Answer:

(-5, -3)

Explanation:

We can use the fact that the diagonals of a parallelogram bisect each other

Let me classify the points as

A (-4, 7)

B (2, -5)

C (3, 5)

Let us assume the 4th vertex is D(x, y) where x and y have to be determined

We will assume that AB is the diagonal of the parallelogram ABCD

The midpoint of a line segment from (x1, y1) to (x2, y2) is given by
(1/2)(x1 + x2, y1 + y2) ie the average of their x and coordinates

If we let O be the midpoint of AB, coordinates of O are
1/2 x (-4 + 2, 7 -5) = 1/2 x ( -2, 2) = (-1, 1)

We know the other diagonal CD must pass through point O and O is also the midpoint of CD

Therefore if coordinates of D are (x, y)

then x-coordinate of O which is -1 should be the average of x coordinate of C and x coordinate of D
==> - 1 = (3 + x)/2

=> -2 = 3 + x

=> -2 - 3 = x

=> -5 = x

=> x = -5

Similarly
1 = (5 + y)/2

=> 2 = 5 + y

=> 2 - 5 = y

=> -3 = y

y = -3

So the coordinate of the fourth vertex D are (-3, -5)

Check the attached graphs for a clearer understanding

Three vertices of a parallelogram are shown in the figure below. Give the coordinates-example-1
Three vertices of a parallelogram are shown in the figure below. Give the coordinates-example-2
User Jo Smo
by
3.9k points