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The perimeter of a rectangle and an equilateral triangle are same. Also one of the sides of the rectangle is equal to the side of the triangle. The ratio of the area of the rectangle and the triangle is
Answer & Solution
Correct Answer:
Option
C
2($$l$$ + b) = 3a
(a = side of equilateral triangle)
Let (b = a)
⇒ 2($$l$$ + a) = 3a
⇒ 2$$l$$ + 2a = 3a
⇒2$$l$$ = a
⇒ $$l$$ = $$\frac{{\text{a}}}{2}$$
Required Ratio
$$\eqalign{ & = \frac{{l \times b}}{{\frac{{\sqrt 3 }}{4}{a^2}}} \cr & = \frac{{\frac{a}{2} \times a}}{{\frac{{\sqrt 3 }}{4}{a^2}}} \cr & = \frac{{{a^2}}}{2} \times \frac{4}{{\sqrt 3 {a^2}}} \cr & = \frac{2}{{\sqrt 3 }} \cr & = {\bf{2:}}\sqrt {\bf{3}} \cr} $$
(a = side of equilateral triangle)
Let (b = a)
⇒ 2($$l$$ + a) = 3a
⇒ 2$$l$$ + 2a = 3a
⇒2$$l$$ = a
⇒ $$l$$ = $$\frac{{\text{a}}}{2}$$
Required Ratio
$$\eqalign{ & = \frac{{l \times b}}{{\frac{{\sqrt 3 }}{4}{a^2}}} \cr & = \frac{{\frac{a}{2} \times a}}{{\frac{{\sqrt 3 }}{4}{a^2}}} \cr & = \frac{{{a^2}}}{2} \times \frac{4}{{\sqrt 3 {a^2}}} \cr & = \frac{2}{{\sqrt 3 }} \cr & = {\bf{2:}}\sqrt {\bf{3}} \cr} $$
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