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

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Solution

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

Using the above theorem prove that a line through the point of intersection of the diagonals and parallel to the base of the trapezium divides the non parallel sides in the same ratio.

Ans.
For the Theorem :

Given, To prove, Construction and figure of 1½ marks

Proof of 1½ marks

Let ABCD be a trapezium DC $∥$ AB and EF is a line parallel to AB and hence to DC.

Join AC, meeting EF in G.

In $∆ A B C$ , we have $G F ∥ A B$

$\frac{C G}{G A} = \frac{C F}{F B} [By BPT]$ ...(1)

In $∆ A D C$ , we have $E G ∥ D C$ ( $E F ∥ A B & A B ∥ D C$ )

$\frac{D E}{E A} = \frac{C G}{G A} [By BPT]$ ...(2)

From (1) & (2), we get, $\frac{D E}{E A} = \frac{C F}{F B}$

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