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CAT 2025Slot 1QAQues & Sol

GeometryHard

Question

If the length of a side of a rhombus is 36 cm and the area of the rhombus is 396 sq. cm, then the absolute value of the difference between the lengths, in cm, of the diagonals of the rhombus is

Solution

1. Concept Used

  • Topic: Geometry – Properties of a Rhombus (Diagonals and Area)
  • Formula: $$\text{Area} = \frac{1}{2} d_1 d_2 \quad \text{and} \quad \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2$$

2. Calculation

Let the two diagonals of the rhombus be $d_1$ and $d_2$, and the side length be $a = 36$ cm.

Using the area formula: $$\frac{1}{2} d_1 d_2 = 396 \implies d_1 d_2 = 792$$

The diagonals of a rhombus bisect each other at right angles, forming four right-angled triangles, each with legs $\frac{d_1}{2}$ and $\frac{d_2}{2}$, and hypotenuse equal to the side $a$. Applying the Pythagorean theorem: $$\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2 \implies \frac{d_1^2 + d_2^2}{4} = 36^2 = 1296 \implies d_1^2 + d_2^2 = 5184$$

Now, to find $|d_1 - d_2|$, we use the algebraic identity: $$(d_1 - d_2)^2 = d_1^2 + d_2^2 - 2d_1 d_2$$

Substituting the known values: $$(d_1 - d_2)^2 = 5184 - 2 \times 792 = 5184 - 1584 = 3600$$

Taking the square root: $$|d_1 - d_2| = \sqrt{3600} = 60$$

3. Solution

Answer = 60 ✅

The absolute value of the difference between the lengths of the diagonals of the rhombus is 60 cm.