CAT 2025 Slot1 Quant: If the length of a side of a rhombus is 36 cm and the area of the rhombus is 396 sq. cm

Step 1: Formulae for Rhombus
Let the diagonals be $d_1$ and $d_2$ and the side be $a = 36$.
1. Area $A = \frac{1}{2} d_1 d_2 = 396 \implies d_1 d_2 = 792$.
2. Relation between side and diagonals: $a^2 = (\frac{d_1}{2})^2 + (\frac{d_2}{2})^2 \implies 4a^2 = d_1^2 + d_2^2$.
$d_1^2 + d_2^2 = 4 \times 36^2 = 4 \times 1296 = 5184$.

Step 2: Find the difference between diagonals
We need $|d_1 - d_2|$. We know that:
$(d_1 - d_2)^2 = d_1^2 + d_2^2 - 2d_1 d_2$.
Substitute the values:
$(d_1 - d_2)^2 = 5184 - 2(792)$.
$(d_1 - d_2)^2 = 5184 - 1584 = 3600$.

Step 3: Calculate the square root
$|d_1 - d_2| = \sqrt{3600} = \mathbf{60}$.
The absolute difference between the diagonals is 60 cm.