wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

D, E and F are respectively the mid-points of the sides BC, CA and AB of a ABC. Show that
(i) BDEF is a parallelogram.(ii) ar (DEF) = 14 ar (ABC). [4 MARKS]

Open in App
Solution

Statement of the theorem : 1 mark
Application of the theorem : 1 mark
Calculation : 2 marks

Given: D, E and F are respectively the mid-points of the sides BC, CA and AB of ABC.

i.) In ABC, F is the midpoint of AB and E is the midpoint of AC

Therefore, EF BD---------(1) [Converse of mid-point theorem]

Similarly, BF DE----------(2)

In view of (1) and (2), BDEF is a parallelogram.


ii.) As in (i) , we can prove that AFDE and FDCE are parallelograms.

FD is the diagonal of the parallelogram BDEF.

So, ar(FBD) = ar(DEF)----(3)

Similarly, ar(DEF) = ar(FAE)-----(4)

And ar(DEF) = ar(DCE)----(5)

From (3), (4) and (5)

ABC is divided into 4 triangles of equal areas.


Therefore, ar(ABC) = ar(FBD) + ar(DEF) + ar(FAE) + ar(DCE)

= 4 × ar(DEF)


ar(DEF) = 14ar(ABC)


flag
Suggest Corrections
thumbs-up
39
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Parallelograms
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon