Question

Given a semicircle of radius $1$, let $a$ be the side of an equilateral triangle which is inscribed in the semicircle with its vertices on the boundary of the semicircle (boundary includes the bounding diameter also). Then the set of possible values of $a$ is

• A. $\left\{1\right\}$
• B. $\{1,\frac{2}{\sqrt{3}}\}$
• C. the set of all positive real numbers not exceeding $\frac{2}{\sqrt{3}}$
• D. the set of all real numbers which are greater than or equal to 1, but less than or equal to $\frac{2}{\sqrt{3}}$

Answer 1

Answer: (B)

$\{1,\frac{2}{\sqrt{3}}\}$

For $a$ to be the side of triangle

Height of triangle is $\frac{a}{2}\tan {60}^o=\frac{\sqrt{3}a}{2}$

For $a=1,h=\frac{\sqrt{3}}{2}$ lies on boundary

For $a=\frac{2}{\sqrt{3}},h=1$ lies on boundary

For any other value of $0<a<1$ other than $\frac{2}{\sqrt{3}}$ the triangle lies inside the semicircle

For any value of $a>1$ the triangle crosses the semicircle.