Question

XY is a line parallel to side BC of a triangle ABC. If BE || CA and FC || AB meet XY at E and F respectively, show that area of ∆ABE = area of ∆ACF.

Answer 1

Answer:

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Answer 2

area of ∆ABE = area of ∆ACF

Explanation:

Given: XY is a line parallel to side BC of a triangle ABC.  

BE || CA and FC || AB meet XY at E and F respectively

To prove: area of ∆ABE = area of ∆ACF.

Proof: Let XY intersect AB and BC at M and N respectively.

As XY || BC,  then EN || BC.

Also BE || AC, then BE || CN.

BCNE is a parallelogram as the opposite sides are parallel.

Parallelogram BCNE and ∆AEB lie on the same base BE, and lie between same parallel lines BE and AC.

So area of ∆AEB = 1/2 area of BCNE ----------------------(1)

Similarly, area of ∆ACF = 1/2 area of BCFM ----------------------(2)

Parallellograms BCNE and BCFM are on the same base BC and lie between the same parallel lines BC and EF.

So area of BCNE = area of BCFM ----------------------(3)

From (1), (2) and (30), we get:

area of ∆ABE = area of ∆ACF

Hence proved.