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in Perimeter and Area of Plane Figures by (23.5k points)
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If the distance from the vertex to the centroid of an equilateral triangle is 6 cm, then what is the area of the triangle?

(a) 24 cm

(b) 27√3 cm

(c) 12 cm

(d) 12√3 cm

1 Answer

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(b) 27√3 cm2.

Let G be the centroid of ΔPQR. Then, PG = 6 cm.

Now, \(\frac{PG}{GS}\) = \(\frac{2}{1}\) ⇒ GS = 3 cm

∴PS = PG + GS = 9 cm            …(i)

∴ If a is the length of a side of ΔPQR, then ΔPQR being equilateral, PS⊥QR

∴ Altitude PS = \(\frac{\sqrt3}{2}\) a = 9 (From (i))

⇒ a = \(\frac{9\times2}{\sqrt3}\) = 6√3 cm

∴ Area of equilateral ΔPQR = \(\frac{\sqrt3}{4}\) (a)2

\(\frac{\sqrt3}{4}\) x (6√3)cm2 = 27√3 cm2.

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