The length of three medians of a triangle are 9 cm, 12 cm and 15 cm. Then the area of triangle is:

Concept:

Area of triangle = \(\frac{4}{3}\) ×(Area of the triangle formed by median as a side) 

The area of a triangle whose side lengths are a, b and c is given by:

\(\rm A = \sqrt{s(s-a)(s-b)(s-c)}\), Where 's' is semi-perimeter of the triangle.

Semi-perimeter of the triangle = s = \(\rm \frac{a+b+c}{2}\)

Calculation:

Given: length of three medians of a triangle are 9 cm, 12 cm and 15 cm

Let s be semi-perimeter of the triangle formed by median as a side

∴ s = \(\rm \frac{9+12+15}{2} = 18\)

Now, Area of the triangle formed by median as a side = \(\rm \sqrt{s(s-a)(s-b)(s-c)}\)

\(=\sqrt{18(18-9)(18-12)(18-15)} \\= \sqrt{18 \times 9\times 6\times 3} \\= 54\)

As we know,

Area of triangle = \(\frac{4}{3}\) ×(Area of the triangle formed by median as a side) 

= \(\rm = \frac{4}{3}\times 54 = 72 cm^2\)