April 13, 2020

Centroids and Area | PRMO 2018 | Question 21

Content

Try this beautiful problem from the PRMO, 2018 based on Centroids and Area.

Centroids and Area - PRMO 2018

Let ABC be an acute angled triangle and let H be its orthocentre. Let \(G_1\),\(G_2\) and \(G_3\) be the centroids of the triangles HBC, HCA, HAB. If area of triangle \(G_1G_2G_3\) =7 units, find area of triangle ABC.

is 107
is 63
is 840
cannot be determined from the given information

Key Concepts

Orthocentre

Centroids

Similarity

Check the Answer

But try the problem first...

Answer: is 63.

Source

Try with Hints

First hint

AB=2DE in triangle \(HG_1G_2\) and triangle \(HDE\) \(\frac{AG_1}{HD}=\frac{G_1G_2}{DE}=\frac{2}{3}\) then \(G_1G_2=\frac{2DE}{3}=\frac{2AB}{3 \times 2}=\frac{AB}{3}\)

Second Hint

triangle \(G_1G_2G_3\) is similar triangle ABC then \(\frac{AreaatriangleABC}{Area G_1G_2G_3}=\frac{AB^{2}}{G_1G_2^{2}}=9\)

Final Step

then area triangle ABC=\(9 \times area triangle G_1G_2G_3\)=(9)(7)=63.