Question

From a triangle ABC with sides of lengths 40 ft, 25 ft and 35 ft, a triangular portion GBC is cut off where G is the centroid of ABC. The area, in sq ft, of the remaining portion of triangle ABC is

• A. $225\sqrt{3}$
• B. $\frac{500}{\sqrt{3}}$
• C. $\frac{275}{\sqrt{3}}$
• D. $\frac{250}{\sqrt{3}}$

Answer 1

Answer: (B)

$\frac{500}{\sqrt{3}}$

Answer 2

The formula for the area of a triangle, given its sides, is expressed as:
Area $=\sqrt{s(s−a)(s−b)(s−c)}​$
where s is the semi-perimeter of the triangle, and a ,b ,c are the lengths of its sides.

In the case of triangle ABC with sides 40,35, and 25, the semi-perimeter s is calculated as:
$s=\frac{40+35+25}{2}​=50$

Substituting these values into the area formula:
Area$=\sqrt{50×10×15×25}​=250\sqrt{3}​$

Since the centroid divides the medians in a 2:1 ratio, the area of triangle GBC is $\frac{1}{3}$​ of the area of triangle ABC :
Area of triangle GBC $=\frac{1}{3}×$Area of triangle ABC
Therefore, the required area is $\frac{2}{3}$​ times the area of triangle ABC :

Required Area$=\frac{2}{3}×250\sqrt{3}=\frac{500}{\sqrt{3}}$