Question

In the equilateral triangle ABC, AD = DE = BE, D and E lies on the AB. If each side of the triangle (i.e., AB, BC and AC) be 6 cm, then the area of the shaded region is :

• A. $9c{m}^2$
• B. $6\sqrt{3}c{m}^2$
• C. $5\sqrt{3}c{m}^2$
• D. $Noneoftheabove$

Answer 1

The correct option is B $6\sqrt{3}c{m}^2$

Area of equilateral triangle
$ABC=\frac{\sqrt{3}}{4}\times (6{)}^2=9\sqrt{3}c{m}^2$
Area of $\Delta CDE=\frac{1}{2}\times DE\times CP$
$=\frac{1}{2}\times 2\times \frac{\sqrt{3}}{2}\times 6$
$=3\sqrt{3}c{m}^2$
Hence, Area of the shaded region = $9\sqrt{3}c{m}^2-3\sqrt{3}c{m}^2$ = $6\sqrt{3}c{m}^2$

Alternatively,
Since, AD = DE = BE => Area of shaded region = $\frac{2}{3}\times \frac{\sqrt{3}}{4}\times (6{)}^2$
= $6\sqrt{3}c{m}^2$