Question

D,E,F ARE THE MIDPIONT OF THE SIDES BC, CA, AND AB OF A TRIANGIE ABC PROVE THAT BDEF IS A Parallelogram, area(def) =1/4area(abc),area(bdef)=1/2(abc)

Answer 1

Hello Mate!

Given : D,E,F ARE the midpoint of the sides BC, CA, AND AB of ∆ABC.

To prove : (i) BDEF is ||gm
(ii) ar(∆DEF) = ¼ ar(∆ABC)
(iii) ar(∆BDEF) = ½ ar(∆ABC)

Proof : Since EF is formed by mid points soz

EF || BC and EF = ½ BC

EF || BD and EF = BD

Hence, BDEF is ||gm. [ Hence proved (i) ]

Now, similarly CDFE and AFDE is ||gm.

In BDEF, ar(∆BDF) = ar(∆DEF)

In CDFE, ar(∆CDE) = ar(∆DEF)

In AFDE, ar(∆AFE) = ar(∆DEF )

[ Because diagonal divide ||gm in 2 equal parts ]

So, ar(∆DEF) = ar(∆BDF) = ar(∆CDE) = ar(∆AFE)

ar(∆DEF) + ar(∆BDF) + ar(∆CDE) + ar(∆AFE) = ar(∆ABC)

ar(∆DEF) + ar(∆DEF) + ar(∆DEF) + ar(∆DEF) = ar(∆ABC)

4ar(∆DEF) = ar(∆ABC)

ar(∆DEF) = ¼ ar(∆ABC) [ Hence proved (ii) ]

½ ar(||gm BDEF) = ¼ ar(∆ABC)

ar(||gm BDEF) = ½ ar(∆ABC) [ Hence proved (iii) ]

"Q.E.D"

Have great future ahead!

Answer 2

HEY MATE HERE IS YOUR ANSWER

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Note :- In these type of questions we should start with "Given, To prove, Proof". Because these type of questions will come in 3 or 4 marks.


Solution :-

Given :-

D, E, F are mid point of sides BC, CA AND AB.

To prove :-

1. BDEF is ||gm
2. ar ⛛ DEF = 1/4 ar ⛛ ABC
3. ar ⛛ BDEF = 1/2 ar ⛛ ABC

Proof :


Since EF is formed by mid point by so,

➡️ EF || BC and EF = 1/2 BC - - - - - (1)

➡️ EF || AD and EF = BD----------(1)

Therefore, BY EQUATIONS (1) AND (2)

BDEF is a ||gm.

Now,

Similarly

CDFE AND ADFE are also ||gm.

In BDEF, (⛛ BDF) = ar(DEF)

In CDFE, (⛛ CDE) = ar(DEF)

➡️ In AFDE ar(AFE) = ar(DEF)

Diagonal ➗ ||gm in 2 parts.

Therefore,


➡️ ar (DEF) = ar ( BDF) = ar (CDE) = ar( AFE)

➡️ AR (DEF) + AR( BDF) + AR ( CDE) +AR (AFE) = AR (ABC)

➡️ AR(DEF) +AR (DEF) + AR(DEF) + AR(DEF) = AR(ABC)

➡️ 4 AR(DEF) = AR(ABC)

➡️ AR (DEF) = 1/4 AR(ABC) ii) Hence proved

➡️ 1/2 ar(||gm BDEF) = 1/4 ar ( ABC)

➡️ AR(||gm BDEF) = 1/2 AR(ABC) iii) Hence proved










Hope it will help you

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