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April 13, 2020

Centroids and Area | PRMO 2018 | Question 21

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.

Centroids and Area- Figure
  • is 107
  • is 63
  • is 840
  • cannot be determined from the given information

Key Concepts


Orthocentre

Centroids

Similarity

Check the Answer


Answer: is 63.

PRMO, 2018, Question 21

Geometry Vol I to IV by Hall and Stevens

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.

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