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Question 16 : The sum of perimeters of an equilateral triangle and a rectanmgle is 90 cm. The area, T, of the triangle and the area , R, of the rectangle, both in sq cm, satisfy the relationship R = T2. If the sides of the rectangle are in the ratio 1 : 3, then the length, in cm, of the longer side of the rectangle, is

  1. 27
  2. 18
  3. 21
  4. 24

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Explanatory Answer

Sides of the rectangle are in the ratio 1 : 3
Take as x and 3x
Area of Rectangle = 3x2 = R
Take an equilateral triangle with side a
Area = \\frac{√3}{4}) × a2 = T
Given R = T2
3x2= \\frac{√3}{4}) × a2
Solving we get a2 = 4x
Now from the options longer side 3x = 27
x = 9. Sides of the rectangle are 9 and 27.
If x = 9, Then a = 6
Checking if a2 = 4x
62 = 4(9)
It works, hence longer side of rectangle is 27

The answer is, "27"

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