The sum of the perimeters of an equilateral triangle and a rectangle 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
Explanation:
Let the side of equilateral triangle be a. Hence, perimeter = 3a.
Let the sides of the rectangle be x and 3x. Hence, perimeter = 8x
Given, 3a + 8x = 90 …(1)
Also given, R = T2
⇒ x × 3x = 34a22
⇒ 16x2 = a4
⇒ a2 = 4x
Substituting x in (1)
⇒ 3a + 2a2 = 90
⇒ 2a2 + 3a – 90 = 0
⇒ a = -15/2 or 6 (-ve value will be rejected)
∴ x = a24 = 9
∴ Longer side of the rectangle = 3x = 27.
Hence, option (a).
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