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

The Correct Option is B

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\(=\frac{2}{3}×250\sqrt{3}=\frac{500}{\sqrt{3}}\)