Q3 of 42 Page 214

D, E, F are mid points of sides BC, CA, AB to Δ ABC. Find the ratio of areas of ΔDEF and Δ ABC.


⇒ Given, D, E, F are mid points of BC, CA, AB


⇒ Need to find the ratios of Δ DEF and Δ ABC


⇒ DE || AF or DF || BE


⇒ similarly EF || AB or EF || DB


⇒ AFED is a parallelogram as both pair of opposite sides are parallel


⇒ By the property of parallelogram


⇒ ∠ DBE = ∠ DFE


Or ∠ DFE = ∠ ABC …………eq(1)


⇒ Similarly ∠ FEB = ∠ ACB …..eq(2)


⇒ In Δ DEF and Δ ABC from eq(1) and eq(2) we have


⇒ Δ DEF ∼ Δ CAB


⇒


⇒ ar(Δ DEF) : ar(Δ ABC) = 1:4


Hence proved


More from this chapter

All 42 →
1

Equilateral triangles are drawn on the three sides of a right angled triangle. Show that the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.

 2

Prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangles described on its diagonal.

 4

In Δ ABC, XY || AC and XY divides the triangle into two parts of equal area. Find the ratio of

5

Prove that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding medians.