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 $\triangle ARC$, $\angle ARC = 90^\circ$, and in $\triangle PQS$, $\angle PQS = 90^\circ$. The length of QS is 6 cm. What is the ratio of $AP : PD$?
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 $\triangle ARC$, $\angle ARC = 90^\circ$, and in $\triangle PQS$, $\angle PQS = 90^\circ$. The length of QS is 6 cm. What is the ratio of $AP : PD$?
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For problems involving similar triangles, use the properties of parallel lines and proportionality to calculate the ratios of the sides.
Updated On: Aug 1, 2025
- 10 : 3
- 2 : 1
- 3 : 7
- 8 : 3
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The Correct Option is D
Solution and Explanation
Given that PQ is parallel to AC and QD is parallel to CP, we can use similar triangles and the properties of parallel lines to determine the ratio of $AP : PD$.
Since $\triangle ARC$ is a right triangle, and $\triangle PQS$ is also a right triangle, by the properties of similar triangles, the sides of these triangles will be proportional.
From the given information, we know that the length of $QS$ is 6 cm. By applying the proportionality of the sides of the triangles and using the ratio of $AP : PB = 4 : 3$, we find that the ratio of $AP : PD$ is 8 : 3.
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