Q) 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.

(Q 29 – 30/2/3 – CBSE 2026 Question Paper)

Ans:

Let’s consider the coordinates of fourth vertex S are (m, n).

Step 1: We know that the in a Rhombus, the diagonals bisect each other.

It means that the midpoint of one diagonal is the same as the midpoint of the other diagonal .

Step 2: Next, let’s calculate midpoint of diagonal PR:

We know that by midpoint formula,

the midpoint of two points (x₁, y₁) and (x₂,y₂) is given by:

(x, y) =

∴ Midpoint of points P (2, 3) and R (- 2, 1):

T (x, y) =

∴ T (x,y) = (0, 2) ……………. (i)

These are the coordinates of the intersection point of both the diagonals.

Step 3: Let’s calculalte midpoint of diagonal QS:

By above mid-section formula, midpoint of points Q (6, 5) and S (m, n): 

T (x, y) =   ……….. (ii)

Step 4: Now, since both these midpoints refer to same point,

Let’s compare x – coordinates from both equations:

∴ = 0

∴ (6 + m) = 0

∴ m = – 6

Step 5: Similarly, we compare y – coordinates from both equations

∴ = 2

∴ (5 + n) = 4

∴ n = 4 – 5 = – 1

Therefore, the coordinates of fourth vertex S are (- 6, – 1).

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