Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

Let A B C be an acute-angled triangle and let H be its orthocentre....

Let ABCA B C be an acute-angled triangle and let HH be its orthocentre. Let G1,G2G_1, G_2 and G3G_3 be the centroids of the triangles HBC,HCA\mathrm{HBC}, \mathrm{HCA} and HAB respectively. If the area of triangle G1G2G3\mathrm{G}_1 \mathrm{G}_2 \mathrm{G}_3 is 7 units, what is the area of triangle ABCA B C ?Let A B C be an acute-angled triangle and let H be its orthocentre. Let G_1, G_2 and G_3 be the centroids of the

Correct answer

63

Step-by-step solution

Let A B C be an acute-angled triangle and let H be its orthocentre. Let G_1, G_2 and G_3 be the centroids of the
AB=2DEA B=2 D E....(i) In ΔHG1G2&ΔHDE\Delta \mathrm{H} \mathrm{G}_1 \mathrm{G}_2 \quad \& \quad \Delta \mathrm{HDE} HG1HD=G1G2DE=23\frac{\mathrm{HG}_1}{\mathrm{HD}}=\frac{\mathrm{G}_1 \mathrm{G}_2}{\mathrm{DE}}=\frac{2}{3} G1G2=23DE=23(AB2)=AB3\mathrm{G}_1 \mathrm{G}_2=\frac{2}{3} \mathrm{DE}=\frac{2}{3}\left(\frac{\mathrm{AB}}{2}\right)=\frac{\mathrm{AB}}{3} Becomes ΔG1G2G3∼ΔABC\Delta \mathrm{G}_1 \mathrm{G}_2 \mathrm{G}_3 \sim \Delta \mathrm{ABC} ⇒ Area of ΔABC Area of ΔG1G2G3=(AB)2(G1G2)2=(ABG1G2)2=(31)2\Rightarrow \frac{\text { Area of } \Delta A B C}{\text { Area of } \Delta G_1 G_2 G_3}=\frac{(A B)^2}{\left(G_1 G_2\right)^2}=\left(\frac{A B}{G_1 G_2}\right)^2=\left(\frac{3}{1}\right)^2 ⇒\Rightarrow Area of △ABC=9×(\triangle \mathrm{ABC}=9 \times\left(\right. Area of \left.\triangle \mathrm{G}_1 \mathrm{G}_2 \mathrm{G}_3\right)⇒\Rightarrow Area of △ABC=9×7=63\triangle \mathrm{ABC}=9 \times 7=63
Practice Properties of Triangles on Quantrex Academy →

More from Properties of Triangles

The incircle Γ of a scalene triangle A B C touches B C at D, C A at E and A B at F . Let… Olympiad workbookThe sides x and y of a scalene triangle satisfy x+(2 Δ)/(x)=y+(2 Δ)/(y) , where Δ is the… Olympiad workbookIn a triangle A B C , the median A D divides ∠ B A C in the ratio 1: 2 . Extend A D to E… Olympiad workbookLet A B C be a triangle let D be a point on the segment B C such that A D=B C . Suppose ∠… Olympiad workbookTwo sides of a regular polygon with n sides, when extended, meet at an angle of 28^ ∘ .… Olympiad workbookLet A B C be an isosceles triangle with A B=A C and incentre I . If A I=3 and the… Olympiad workbookLet A B C be an acute-angled triangle and P be a point in its interior. Let P_A, P_B… Olympiad workbookLet A B C be a triangle with ∠ B A C=90^ ∘ and D be the point on the side B C such that A… Olympiad workbook Full Properties of Triangles list All Olympiad workbook PYQs