Question

If an equilateral triangle of area $X$ and a square of area $Y$ have the same perimeter, then $X$ is:

• A. equal to $Y$
• B. greater than $Y$
• C. less than $Y$
• D. less than or equal to $Y$

Answer 1

Answer: (C)

less than $Y$

Let each side of triangle be ${}^{\prime }{a}^{\prime }$cm & each side of square be ${}^{\prime }{b}^{\prime }$cm.

$\therefore X=\frac{\sqrt{3}}{4}{a}^2$ and $Y=b^2$

Perimeter of triangle $=3a$ and Perimeter of square $=4b$

Given, $3a=4b\Rightarrow b=\frac{3a}{4}$

$\therefore X=\frac{\sqrt{3}}{4}{a}^2,Y={\left(\frac{3a}{4}\right)}^2=\frac{9a^2}{16}$

Now, $\frac{\sqrt{3}}{4}{a}^2=\frac{1.732}{4}{a}^2=0.433a^2$

And, $\frac{9a^2}{16}=0.5625a^2$

$\therefore X<Y$