Question

prove bpt rule
class 10​

Answer 1

Answer:

Here is your answer

Step-by-step explanation:

BPT theorem :- " If a line is drawn parallel to one side of a triangle interacting the other two sides in distinct points, then the other two sides are divided in the same ratio."

Q. GIVEN:- In the ∆ ABC's BC ll DE .

WHAT TO PROVE :- We have to prove that AD\DB = AE/EC

CONSTRUCTION :- Join D,C and B,E. Draw EG perpendicular AD and DF perpendicular AE.

prove:- Now,

We found in the ∆ADE -----

ar(ADE) = 1/2 × b × h

= 1/2 × AD × EG. ⟨--------- (I)

and ar(ADE) = 1/2 × b × h

=. 1/2 × AE × DF ⟨-------- (ii)

Again,

We found in the ∆BDE and ∆ DEC -----

ar(BDE) = 1/2 × b × h

= 1/2 × DB × EG ⟨---------- (iii)

and , ar(DEC) = 1/2 × b × h

= 1/2× EC × DF⟨---(iv)

Now,

Deviding (I) by (iii)-----

ar(ADE)/ar(BDE)= 1/2×AD×EG/ 1/2×DB×EG

= AD/DB ⟨----- (V)

again,

Deviding (ii) by (iv)-----

ar(ADE)/ar(DEC)= 1/2×AE×DF/ 1/2×EC×DF

= AE/EC ------(VI)

We know that , ∆BDE and ∆DEC's base is same and DE||BC.

Therefore,

ar(BDE)=ar(DEC) ---(vii)

From (v),(vi) and (vii) we get-----

AD/DB = AE/EC.

PROVED .

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