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CAT 2025 Slot 1 QA Question 12

Type-in-the-answer (no negative marking) · Geometry · Quadrilaterals · Try it, then check the answer and solution below.

CAT 2025 Slot 1QAGeometry • QuadrilateralsModerate
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
TITA Answer:
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Answer and solution

Answer: 60

📌 Core Concept
The area of a rhombus can be calculated using the formula: $Area=d1×d22\text{Area} = \frac{d_1 \times d_2}{2} where d1d_1 and d2d_2 are the lengths of the diagonals.
🔢 Step-by-Step Solution
1
Given:
Side length of the rhombus, a=36a = 36 \text{ cm}.
Area of the rhombus, A=396A = 396 sq. \text{ cm}.
2
Using the area formula:
d1×d22=396  ⟹  d1×d2=792\frac{d_1 \times d_2}{2} = 396 \implies d_1 \times d_2 = 792
3
Using the Pythagorean theorem:
The diagonals bisect each other at right angles, forming right-angled triangles with legs d12\frac{d_1}{2}andd22,andhypotenuseand \frac{d_2}{2} , and hypotenusea = 36$ \text{ cm}.
(d12)2+(d22)2=362\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = 36^2
Simplifying:
d124+d224=1296  ⟹  d12+d22=5184\frac{d_1^2}{4} + \frac{d_2^2}{4} = 1296 \implies d_1^2 + d_2^2 = 5184
4
Finding ∣d1−d2∣|d_1 - d_2|:
Use the identity (d1−d2)2=d12+d22−2d1d2(d_1 - d_2)^2 = d_1^2 + d_2^2 - 2d_1d_2.
(d1−d2)2=5184−2×792=5184−1584=3600(d_1 - d_2)^2 = 5184 - 2 \times 792 = 5184 - 1584 = 3600
Taking the square root: $$
|d_1 - d_2| = \
3600\sqrt{3600}
= 60 $$
⚡ 30-Second Shortcut
Recognize that the area formula and the Pythagorean theorem with the side length give two equations. Use the identity for the square of the difference of diagonals to find the answer quickly.
🎯 Final Answer Confirmation
Correct Answer: 60

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