In $Δ P Q R$ , if PS is the bisector of $∠ Q P R$ , then which of the following is always true?

P Q \times Q S = P R \times S R

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\frac{P Q}{P R} = \frac{Q S}{S R}

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\frac{P Q}{P S} = \frac{P S}{P R}

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\frac{P S}{P Q} = \frac{Q S}{S R}

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Solution

The correct option is B $\frac{P Q}{P R} = \frac{Q S}{S R}$

Draw $R T | | P S$ .

In $Δ Q R T,$
$\frac{Q S}{S R} = \frac{Q P}{P T}$ (BPT) ......(i)
Since, PR is transversal,
$∴ ∠ S P R = ∠ P R T$ (Alternate interior angles) ..… (ii)
and $∠ Q P S = ∠ P T R$ (Corresponding angles) ..… (iii)
Given that PS is bisector of $∠ Q P R$ .
$\Rightarrow ∠ Q P S = ∠ R P S$ ..… (iv)
$\Rightarrow ∠ P T R = ∠ P R T$ [From (ii), (iii) and (iv)
$\Rightarrow P T = P R$ (Sides opposite to equal angles are equal)
From (i), $\frac{P Q}{P R} = \frac{Q S}{S R}$
Hence, the correct answer is option (b).

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