Question

State the basic proportionality theorem. Use it to prove the following:

If three parallel lines l, m, n are intersected by transversals q and s as shown in the adjoining figure, then `(AB)/(BC) = (DE)/(EF)`.

Answer 1

If a line is drawn parallel to one side of a triangle and intersects the other two sides at different points, it divides those sides in the same ratio.

Draw a line segment intersecting line m at point G.

In ΔACF, BE || CF

`(AB)/(BC) = (AG)/(GF)`   ...(1)

In ΔAFD, GE || AD

`(AG)/(GF) = (DE)/(EF)`   ...(2)

From equations (1) and (2), we get

`(AB)/(BC) = (AG)/(GF) = (DE)/(EF)`

⇒ `(AB)/(BC) = (DE)/(EF)`

Hence proved.