# Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two sides are divided in the same ratio. - Mathematics

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Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two sides are divided in the same ratio.

#### Solution

Let a ΔABC in which a line DE parallel to SC intersects AB at D and AC at E.

To prove DE divides the two sides in the same ratio.

i.e, (AD)/(DB) = (AE)/(EC)

Construction: Join BE, CD and draw EF ⊥ AB and DG ⊥ AC.

Proof: Here (ar(ΔADE))/(ar(ΔBDE)) = (1/2 xx AD xx EF)/(1/2 xx DB xx EF)   .......[∵ Area of triangle = 1/2 × base × height]

= (AD)/(DB)  .......(i)

Similarly, (ar(ΔADE))/(ar(ΔDE)) = (1/2 xx AE xx GD)/(1/2 xx EC xx GD) = (AE)/(EG)   ......(ii)

Now, since, ΔBDE and ΔDEC lie between the same parallel DE and BC and on the same base DE.

So, ar(ΔBDE) = ar(ΔDEC)  ......(iii)

From equations (i), (ii) and (iii),

(AD)/(DB) = (AE)/(EC)

Hence proved.

Concept: Similarity of Triangles
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#### APPEARS IN

NCERT Mathematics Exemplar Class 10
Chapter 6 Triangles
Exercise 6.4 | Q 3 | Page 73
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