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.

Hint:

The property of diagonals bisecting each other is true for all parallelograms, including rectangles, rhombuses, and squares.

Answer 1

Step 1: Find Midpoint of Diagonal PR
Given:
P(2, −3)
R(−2, 1)

Using midpoint formula:
M = ((2 + (−2))/2 , (−3 + 1)/2)
= (0/2 , −2/2)
= (0 , −1)

So intersection point M = (0, −1)

Step 2: Use Midpoint Property on QS
In a rhombus, diagonals bisect each other.
So M is also midpoint of QS.

Given Q(6, 5)
Let S(x, y)

Using midpoint formula:
(6 + x)/2 = 0
6 + x = 0
x = −6

(5 + y)/2 = −1
5 + y = −2
y = −7

Final Answer:
Intersection point = (0, −1)
Fourth vertex S = (−6, −7)