Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
Olympiad workbookIOQMProperties of Triangles

Let A B C be an acute-angled triangle and let H be its orthocentre. Let G₁, G₂ and G₃ be the centroids of the triangles HBC , HCA and HAB respectively. If the area of triangle G ₁ G ₂ G ₃ is 7 units, what is the area of triangle A B C ?

Correct answer

63

Step-by-step solution

A B=2 D E ....(i) In H G ₁ G ₂ & HDE HG ₁ HD = G ₁ G ₂ DE = 2 3 G ₁ G ₂= 2 3 DE = 2 3 ( AB 2 )= AB 3 Becomes G ₁ G ₂ G ₃ ABC Area of A B C Area of G₁ G₂ G₃ = (A B)^2 (G₁ G₂ )^2 = ( A B G₁ G₂ )^2= ( 3 1 )^2 Area of ABC =9 ( . Area of . G ₁ G ₂ G ₃ ) Area of ABC =9 7=63

Practice Properties of Triangles on Quantrex Academy →

More from Properties of Triangles

The incircle of a scalene triangle A B C touches B C at D, C A at E and A B at F . Let r_A be the radius of the circle inside A B C which is tangent to and the sides A B and A C . The sides x and y of a scalene triangle satisfy x+ 2 x =y+ 2 y , where is the area of the triangle. If x=60, y=63 , what is the length of the largest side of the triangle?In a triangle A B C , the median A D divides B A C in the ratio 1: 2 . Extend A D to E such that E B is perpendicular A B . Given that B E=3, B A=4 , find the integer nearest to B Let A B C be a triangle let D be a point on the segment B C such that A D=B C . Suppose C A D=x^ , A B C=y^ and A C B=z^ and x, y, z are in an arithmetic progression in that order Two sides of a regular polygon with n sides, when extended, meet at an angle of 28^ . What is the smallest possible value of n ?Let A B C be an isosceles triangle with A B=A C and incentre I . If A I=3 and the distance from I to B C is 2 , what is the square of the length of B C ?Let A B C be an acute-angled triangle and P be a point in its interior. Let P_A, P_B , and P_C be the images of P under reflection in the sides B C, C A and A B , respectively. If Let A B C be a triangle with B A C=90^ and D be the point on the side B C such that A D B C . Let r, r₁ , and r₂ be the inradii of triangles A B C, A B D and A C D , respectively. Full Properties of Triangles list All Olympiad workbook PYQs