Question:
In the figure (not drawn to scale) given below, P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC and QD is parallel to CP. In \( \Delta \)ARC, \( \angle \)ARC = 90\( ^\circ \), and in \( \Delta \)PQS, \( \angle \)PSQ = 90\( ^\circ \). The length of QS is 6 cm. What is the ratio AP : PD?
In the figure (not drawn to scale) given below, P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC and QD is parallel to CP. In \( \Delta \)ARC, \( \angle \)ARC = 90\( ^\circ \), and in \( \Delta \)PQS, \( \angle \)PSQ = 90\( ^\circ \). The length of QS is 6 cm. What is the ratio AP : PD?
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Use the basic proportionality theorem twice: first with PQ parallel to AC in triangle ABC, then with QD parallel to CP in triangle BCP. The right angles and the length QS are not needed.
Updated On: Nov 6, 2026
- 10 : 3
- 2 : 1
- 8 : 3
- 7 : 3
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The Correct Option is D
Solution and Explanation
Step 1: Set up:
Let AP = 4k and PB = 3k, so AB = 7k. Q lies on BC and D lies on AB.
Step 2: PQ || AC:
In \( \Delta \)ABC, PQ is parallel to AC, so \( \dfrac{BQ}{BC} = \dfrac{BP}{BA} = \dfrac{3k}{7k} = \dfrac{3}{7} \).
Step 3: QD || CP:
In \( \Delta \)BCP, QD is parallel to CP, so \( \dfrac{BD}{BP} = \dfrac{BQ}{BC} = \dfrac{3}{7} \). Hence \( BD = \dfrac{3}{7}\times 3k = \dfrac{9k}{7} \).
Step 4: Find PD:
\( PD = PB - BD = 3k - \dfrac{9k}{7} = \dfrac{12k}{7} \).
Step 5: Ratio:
\( AP : PD = 4k : \dfrac{12k}{7} = 28 : 12 = 7 : 3 \).
Final Answer:
The correct option is (4): 7 : 3
Let AP = 4k and PB = 3k, so AB = 7k. Q lies on BC and D lies on AB.
Step 2: PQ || AC:
In \( \Delta \)ABC, PQ is parallel to AC, so \( \dfrac{BQ}{BC} = \dfrac{BP}{BA} = \dfrac{3k}{7k} = \dfrac{3}{7} \).
Step 3: QD || CP:
In \( \Delta \)BCP, QD is parallel to CP, so \( \dfrac{BD}{BP} = \dfrac{BQ}{BC} = \dfrac{3}{7} \). Hence \( BD = \dfrac{3}{7}\times 3k = \dfrac{9k}{7} \).
Step 4: Find PD:
\( PD = PB - BD = 3k - \dfrac{9k}{7} = \dfrac{12k}{7} \).
Step 5: Ratio:
\( AP : PD = 4k : \dfrac{12k}{7} = 28 : 12 = 7 : 3 \).
Final Answer:
The correct option is (4): 7 : 3
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