# If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio - Mathematics

Sum

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio

#### Solution

Let ABC be a triangle in which a line DE is parallel to BC. It intersects the sides AB and AC at D and E respectively.

It has to be proved that,

"AD"/"DB" = "AE"/"EC"

Let us join BE and CD and draw perpendiculars DM and EN on AC and AB respectively.

Area of ΔADE = 1/2 xx "Base" xx "Height"

 = 1/2 xx "AD" xx "EN" = 1/2 xx "AE" xx "DM"

Similarity, ar(ΔBDE) = 1/2 xx "BD" xx "EN"

ar (ΔDEC) 1/2 xx "EC" xx "DM"

("ar"(triangle"ADE"))/("ar"(triangle"BDE"))  =(1/2xx"AD" xx "EN")/(1/2 xx "BD" xx "EN") = "AD"/"BD".....(1)

And, ("ar"(triangle"ADE"))/("ar"(triangle"DEC"))  =(1/2xx"AE" xx "DM")/(1/2 xx "EC" xx "DM") = "AE"/"EC".....(2)

ΔBDE and ΔDEC are on the same base and between the same parallels.

∴ ar (ΔBDE) = ar (ΔDEC)
From (1) and (2), we obtain

"AD"/"BD" ="AE"/"EC"

Concept: Similarity of Triangles
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