Question : In an isosceles triangle, the length of each equal side is twice the length of the third side. The ratios of areas of the isosceles triangle and an equilateral triangle with the same perimeter are:
Option 1: $30\sqrt{5}:{100}$
Option 2: $32\sqrt{5}:{100}$
Option 3: $36\sqrt{5}:{100}$
Option 4: $42\sqrt{5}:{100}$
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Correct Answer: $36\sqrt{5}:{100}$
Solution :
Given: In an isosceles triangle, the length of each equal side is twice the length of the third side.
the third side.
Let the sides of the isosceles triangle be $x, 2x, 2x$, and the sides of the equilateral triangle be $y$ each.
Perimeter of isosceles triangle = Perimeter of equilateral triangle
⇒ $5x = 3y$
⇒ $y = \frac{5}{3}x$
Semi perimeter(s) $=\frac{5x}{2}$
Using Heron's formula, the area of the isosceles triangle is given as,
Area of isosceles triangle
$=\sqrt{[s(s – a)(s – b)(s – c)]}$
$=\sqrt{[\frac{5x}{2}(\frac{5x}{2} – 2x)(\frac{5x}{2} – 2x)(\frac{5x}{2} – x)]}$
$=\sqrt{[\frac{5x}{2}(\frac{x}{2})(\frac{x}{2})(\frac{3x}{2})]}$
$=\frac{\sqrt{15}x^{2}}{4}$
Area of equilateral triangle $=\frac{\sqrt{3}}{4}y^{2}$
Now, the ratio of the area of the isosceles triangle to the area of the equilateral triangle is given as,
$\frac{\frac{\sqrt{15}x^{2}}{4}}{\frac{\sqrt{3}}{4}y^{2}} = \frac{\sqrt{5}x^{2}}{y^{2}} = \frac{9\sqrt{5}}{25}=\frac{9\sqrt{5}}{25} × 4 = \frac{36\sqrt{5}}{100}$
Hence, the correct answer is $36\sqrt{5}:{100}$.
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