Q: 5    D, E and F are respectively the mid-points of the sides BC, CA and AB of a \small \Delta ABC. Show that
           (ii)   \small ar(DEF)=\frac{1}{4}ar(ABC)

Answers (1)


We already proved that BDEF is a ||gm.
Similarly, DCEF and DEAF are also parallelograms.
Now, ||gm BDEF and ||gm DCEF is on the same base EF and between same parallels BC and EF 
\therefore  Ar (BDEF) = Ar (DCEF)
\Rightarrow Ar(\DeltaBDF) = Ar (\DeltaDEF) .............(i)

It is known that diagonals of ||gm divides it into two triangles of equal area.
\therefore Ar(DCE) = Ar (DEF).......(ii)

and, Ar(\DeltaAEF) = Ar (\DeltaDEF)...........(iii)

From equation(i), (ii) and (iii), we get

 Ar(\DeltaBDF) =  Ar(DCE) = Ar(\DeltaAEF) =  Ar (\DeltaDEF)

Thus, Ar (\DeltaABC) =  Ar(\DeltaBDF) +  Ar(DCE) + Ar(\DeltaAEF) +  Ar (\DeltaDEF)
           Ar (\DeltaABC) = 4 . Ar(\DeltaDEF)
         \Rightarrow ar(\Delta DEF) = \frac{1}{4}ar(\Delta ABC)

Hence proved.

Most Viewed Questions

Related Chapters

Preparation Products

Knockout NEET 2024

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 40000/-
Buy Now
Knockout NEET 2025

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 45000/-
Buy Now
NEET Foundation + Knockout NEET 2024

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 54999/- ₹ 42499/-
Buy Now
NEET Foundation + Knockout NEET 2024 (Easy Installment)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 3999/-
Buy Now
NEET Foundation + Knockout NEET 2025 (Easy Installment)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 3999/-
Buy Now
Boost your Preparation for JEE Main with our Foundation Course
 
Exams
Articles
Questions