In the given figure, ABC is an equilateral triangle which is inscribed inside a circle and whose radius is r. Which of the following is the area of the triangle ?

In the given figure, ABC is an equilateral triangle which is inscribed inside a circle and whose radius is r. Which of the following is the area of the triangle ?

Answer: Option D

Solution

Answer: Option D

Solution:

We have :
$$\begin{array}{rl}& AE\perp BC\text{ and }\\ & AD=BD=CD=r\\ & AE=AD+DE=r+DE\end{array}$$
In Δ BDC,
$$\begin{array}{rl}& BE=\sqrt{{\left(BD\right)}^2-{\left(DE\right)}^2}\\ & =\sqrt{r^2-{\left(DE\right)}^2}\\ & =\sqrt{\left(r-DE\right)\left(r+DE\right)}\end{array}$$
∴ Area of the triangle :
$$\begin{array}{rl}& =\frac{1}{2}\times BC\times AE\\ & =\frac{1}{2}\times 2BE\times AE\\ & =BE\times AE\\ & =\sqrt{\left(r-DE\right)\left(r+DE\right)}\left(r+DE\right)\\ & ={\left(r-DE\right)}^{\frac{1}{2}}{\left(r+DE\right)}^{\frac{3}{2}}\end{array}$$