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Basic Proportionality Theorem

In the given ...

Question

In the given figure, $A B | | D E a n d B D | | E F . T h e n, D C^{2} = ?$

C F \times A C

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C F \div A C

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D F \times E B

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D F \div E B

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Solution

The correct option is A
$C F \times A C$
Given : $I n Δ A B C, A B | | D E a n d B D | | E F$

To prove : $D C^{2} = C F \times A C$

Proof : $I n Δ B C D, E F | | B D (G i v e n) \frac{C F}{D C} = \frac{C E}{C B} . . . . (1) (By corollary of BPT) I n Δ A B C, D E | | A B (G i v e n) \frac{D C}{A C} = \frac{C E}{C B} . . . . (2) (By corollary of BPT) F r o m (1) a n d (2), W e g e t \frac{C F}{D C} = \frac{D C}{A C} \Rightarrow C F \times A C = D C \times D C ∴ C F \times A C = D C^{2}$

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