SAT is a triangle. P and Q are points on SA and ST respectively such that PQ || AT. If M is a point on SA such that MQ || PT, then which of the following is always true?

S Q^{2} = S A \times P M

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S P^{2} = S T \times S M

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S P^{2} = S A \times P M

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S P^{2} = S A \times S M

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Solution

The correct option is D $S P^{2} = S A \times S M$
Given: PQ||AT and MQ || PT

Now as PQ||AT,
$\Rightarrow \frac{S P}{S A} = \frac{S Q}{S T}$ (By BPT)....(i)
Since MQ||PT,
$\Rightarrow \frac{S M}{S P} = \frac{S Q}{S T}$ (By BPT)....(ii)
From (i) and (ii),
$\Rightarrow \frac{S P}{S A} = \frac{S M}{S P}$
$\Rightarrow S P^{2} = S A \times S M$
Hence, the correct answer is option (d).

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