The triangle of maximum area that can be inscribed in a given circle of

Question:

The triangle of maximum area that can be inscribed in a given circle of radius ' $r$ ' is:

  1. (1) A right angle triangle having two of its sides of length $2 r$ and $r$.

  2. (2) An equilateral triangle of height $\frac{2 r}{3}$.

  3. (3) An isosceles triangle with base equal to $2 r$.

  4. (4) An equilateral triangle having each of its side of length $\sqrt{3} \mathrm{r}$.


Correct Option: 4,

Solution:

Triangle of maximum area that can be inscribed in a circle is an equilateral triangle. Let $\triangle \mathrm{ABC}$ be inscribed in circle,

Now, in $\triangle O B D O D=r \cos 60^{\circ}=\frac{r}{2}$

Height $=A D=\frac{3 r}{2}$

Again in $\triangle \mathrm{ABD}$ Now $\sin 60^{\circ}=\frac{3 \frac{r}{2}}{A B}$

$\Rightarrow A B=\sqrt{3} r$

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