ISI2015-MMA-32

Question

If a square of side $a$ and an equilateral triangle of side $b$ are inscribed in a circle then $a/b$ equals

• A. $\sqrt{2/3}$
• B. $\sqrt{3/2}$
• C. $3/ \sqrt{2}$
• D. $\sqrt{2}/3$

Answer 1

$\underline{\textbf{Answer: A}}\Rightarrow$

$\underline{\textbf{Explanation:}}\Rightarrow$

When an equilateral triangle is inscribed in a circle then the radius of the circle is given by:

$r = \frac{b}{\sqrt3}$

When a square is inscribed in a circle its diagonal = $\sqrt a$ (Since, side = $a$, hence just use the Pythagoras theorem)

Now, equating both the radius, we get:

$\frac{b}{\sqrt{3}} = \frac{a}{\sqrt 2} = \frac{a}{b} = \sqrt\frac{2}{3}$

Therefore, the correct answer is option $\textbf A$.

Comments

The diagonal of a square with a side length ‘a’ is $\sqrt{2}\text{ }a$.