Ratio of sides of rectangle

CAT 2018 Slot 2 · QA · Easy · Algebra

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The area of a rectangle and the square of its perimeter are in the ratio 1 : 25. Then the lengths of the shorter and longer sides of the rectangle are in the ratio:

  1. A.

    3 : 8

  2. B.

    2 : 9

  3. C.

    1 : 4

  4. D.

    1 : 3

Answer

C

Explanation

Let the sides of the rectangle be aa and bb. Area =ab= ab. Perimeter =2(a+b)  ⟹  = 2(a + b) \implies Square of perimeter =4(a+b)2= 4(a + b)^2.

Given: ab4(a+b)2=125\frac{ab}{4(a+b)^2} = \frac{1}{25} 25ab=4(a2+2ab+b2)25ab = 4(a^2 + 2ab + b^2) 4a2−17ab+4b2=04a^2 - 17ab + 4b^2 = 0 (4a−b)(a−4b)=0(4a - b)(a - 4b) = 0

For a≤ba \le b, b=4a  ⟹  a:b=1:4b = 4a \implies a : b = 1 : 4.

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