The ratio of A P : P D AP:PD

Geometry Level 3

In the figure above (not drawn to scale), P is a point on AB such that A P : P B AP:PB = 4 : 3 4:3 .

Also, P Q ∥ A C PQ \parallel AC and Q D ∥ C P QD \parallel CP .

In △ A R C \triangle ARC , ∠ A R C \angle ARC = 9 0 o 90^{o} . In △ P Q S \triangle PQS , ∠ P S Q \angle PSQ = 9 0 o 90^{o} . The length of Q S QS is 6 cm.

What is the ratio of A P : P D AP:PD ?

2 : 1 2:1 8 : 3 8:3 10 : 3 10:3 7 : 3 7:3 None of these

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1 solution

As P Q ∥ A C PQ \parallel AC ,

C Q Q B \frac{CQ}{QB} = A P P B \frac{AP}{PB} = 4 3 \frac{4}{3}

As Q D ∥ C P QD \parallel CP ,

P D D B \frac{PD}{DB} = C Q Q B \frac{CQ}{QB} = 4 3 \frac{4}{3}

Now, P D D B \frac{PD}{DB} = 4 3 \frac{4}{3}

⟹ \implies P D D B \frac{PD}{DB} + 1 = 4 3 \frac{4}{3} + 1

⟹ \implies P D PD = 4. P B 7 \frac{4.PB}{7}

Thus, A P P D \frac{AP}{PD} = A P ( ( 4. P B ) / 7 ) \frac{AP}{((4.PB)/7)}

= 7 4 \frac{7}{4} x A P P B \frac{AP}{PB} = 7 4 \frac{7}{4} x 4 3 \frac{4}{3} = 7 3 \frac{7}{3}

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