Question

If the distance from the vertex to the centroid of an equilateral triangle is 6cm, what is the area (in cm2) of the triangle?

Answer 1

Equilateral triangle :

Side = a cm.
Altitude = √3/2 a   cm.

Distance from a vertex to the Centroid = 2/3 * √3 /2  * a 
      = a/√3   cm

So  given   a/√3 = 6 cm.
                  a = 6 √3 cm

   Area  = √3/4 * a²
            = √3/4 * (6√3)² = 27√3 cm²

Answer 2

distance from the vertex to the centroid = 6 cm

Let L is the sides of an equilateral triangle.
then, altitude = (√3/2)L

we know,
altitude is divide 2:1 ratio by centroid.
hence, distance from the vertex to the centroid = 2.Altitude/3 = 2(√3/2)L/3 = L/√3 = 6
L = 6√3 cm

now, area of triangle = √3/4(side length)²
= √3/4 × 6√3 × 6√3
= 27√3 cm²