Question

The three vertices of a rhombus PQRS are P(2, –3), Q(6, 5) and R(–2, 1). Find the coordinates of the fourth vertex S and coordinates of the point where both the diagonals PR and QS intersect.

Answer 1

Let the intersection point be M(x, y). It is the midpoint of diagonal PR.

Vertices P(2, –3) and R(–2, 1).

`M = ((2 + (-2))/2, (-3 + 1)/2)`

= `(0/2, (-2)/2)`

= (0, –1)

Now, M(0, –1) is also the midpoint of diagonal QS.

Let S = (x_s, y_s).

Vertex Q(6, 5).

Using the midpoint formula for QS:

`(6 + x_s)/2 = 0`

⇒ 6 + x_s = 0

⇒ x_s = –6

`(5 + y_s)/2 = -1`

⇒ 5 + y_s = –2

⇒ y_s = –7

Coordinates of intersection point are (0, –1) and the fourth vertex S is (–6, –7).