Q1. [1] § 6.3 Similarity of Triangles § 6.4 Criteria for Similarity of Triangles § 6.5 Summary
In the adjoining figure, ABCD is a trapezium in which $XY \parallel AB \parallel CD$. If $AX = \frac{2}{3} AD$, then CY : YB =
- A 2 : 3
- B 3 : 2
- C 1 : 3
- D 1 : 2
Previously asked in CBSE board exam
2025 30/5/1 Q9
Generated by claude-sonnet-4-6 · 2026-06-15 10:26 · grounding rag
Model Answer
Option (D) 1 : 2
Given: AX = 2/3 AD, so XD = AD – 2/3 AD = 1/3 AD.
Thus AX : XD = 2 : 1.
Join diagonal AC (or BD). Since XY ∥ AB ∥ CD, by the Basic Proportionality Theorem applied to the trapezium (using diagonal BD as transversal in △ABD, and diagonal AC in △ABC):
$$\frac{CY}{YB} = \frac{XD}{AX} = \frac{\frac{1}{3}AD}{\frac{2}{3}AD} = \frac{1}{2}$$
∴ CY : YB = 1 : 2
Source: Chapter 6, Section 6.3 (Basic Proportionality Theorem)
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Explanation
- Since AX = 2/3 AD, the remaining part XD = 1/3 AD, giving AX : XD = 2 : 1.
- XY ∥ AB ∥ CD means XY is also ∥ DC. Using the trapezium result (same as Example 2 of the textbook — joining a diagonal and applying BPT), the line XY divides the non-parallel sides AD and BC in the same ratio from the respective vertices.
- On side BC, Y divides it such that CY/YB = XD/AX = 1/2, i.e., CY : YB = 1 : 2.
- Examiners expect you to state BPT, identify the ratio AX : XD correctly, and conclude CY : YB from there.
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