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 ? Options: A . &nbsp$${\left( {r + DE} \right)^{\frac{1}{2}}}{\left( {r - DE} \right)^{\frac{3}{2}}}$$ B . &nbsp$${\left( {r - DE} \right)^{\frac{1}{2}}}{\left( {r + DE} \right)^2}$$ C . &nbsp$${\left( {r - DE} \right)^2}{\left( {r + DE} \right)^2}$$ D . &nbsp$${\left( {r - DE} \right)^{\frac{1}{2}}}{\left( {r + DE} \right)^{\frac{3}{2}}}$$

In the given figure, ABC is an equilateral triangle which is inscribed inside a

Question

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 ?

Options:

A . $${\left( {r + DE} \right)^{\frac{1}{2}}}{\left( {r - DE} \right)^{\frac{3}{2}}}$$

B . $${\left( {r - DE} \right)^{\frac{1}{2}}}{\left( {r + DE} \right)^2}$$

C . $${\left( {r - DE} \right)^2}{\left( {r + DE} \right)^2}$$

D . $${\left( {r - DE} \right)^{\frac{1}{2}}}{\left( {r + DE} \right)^{\frac{3}{2}}}$$

Answer: Option D
We have :
$$\eqalign{
& AE \bot BC{\text{ and }} \cr
& AD = BD = CD = r \cr
& AE = AD + DE = r + DE \cr} $$
In Δ BDC,
$$\eqalign{
& BE = \sqrt {{{\left( {BD} \right)}^2} - {{\left( {DE} \right)}^2}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \sqrt {{r^2} - {{\left( {DE} \right)}^2}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \sqrt {\left( {r - DE} \right)\left( {r + DE} \right)} \cr} $$
∴ Area of the triangle :
$$\eqalign{
& = \frac{1}{2} \times BC \times AE \cr
& = \frac{1}{2} \times 2BE \times AE \cr
& = BE \times AE \cr
& = \sqrt {\left( {r - DE} \right)\left( {r + DE} \right)} \left( {r + DE} \right) \cr
& = {\left( {r - DE} \right)^{\frac{1}{2}}}{\left( {r + DE} \right)^{\frac{3}{2}}} \cr} $$

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