What is the area of \Delta AEC?
Consider the following for the next two (02) items that follow:<br>In the following figure, a rectangle ABCD is inscribed in a circle of radius r. Given \angle DAE=30^{\circ} and \angle ACD=30^{\circ}.
- A. \frac{r^{2}}{\sqrt{3}} ✓
- B. \frac{r^{2}}{2\sqrt{3}}
- C. \frac{r^{2}}{3\sqrt{3}}
- D. \frac{2r^{2}}{\sqrt{3}}
Correct Answer: A. \frac{r^{2}}{\sqrt{3}}
Explanation
In right \triangle ADE, DE = AD \tan 30^{\circ} = \frac{r}{\sqrt{3}}. The base EC = CD - DE = r\sqrt{3} - \frac{r}{\sqrt{3}} = \frac{2r}{\sqrt{3}}. The area of \triangle AEC is \frac{1}{2} \times EC \times AD = \frac{1}{2} \times \frac{2r}{\sqrt{3}} \times r = \frac{r^2}{\sqrt{3}}.
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