Comprehension
These questions are based on the situation given below:
A rectangle PRSU is divided into two smaller rectangles PQTU and QRST by the line TQ. \( PQ = 10 \, \text{cm}, \quad QR = 5 \, \text{cm}, \quad RS = 10 \, \text{cm} \). Points A, B, F are within the rectangle PQTU and, points C, D, E are within the rectangle QRST. The closest pair of points among the pairs \( (A, C), (A, D), (A, E), (F, C), (F, D), (F, E), (B, C), (B, D), (B, E) \) are \( 10\sqrt{3} \, \text{cm} \) apart.
Question: 1
Which of the following statements is necessarily true?
Which of the following statements is necessarily true?
Show Hint
In problems involving distances between points, always calculate the distance between each pair and compare them to determine the closest points.
Updated On: Oct 29, 2026
- The closest pair of points among the six given points cannot be (F, C).
- Distance between A and B is greater than distance between F and C.
- The closest pair of points among 6 given points (C,D), (D,E) or (C,E).
- None of the above.
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The Correct Option is A
Solution and Explanation
Given the coordinates of the points and the distances between them, the closest pair of points is never (F, C), as the distance between them is not the smallest compared to other pairs. Based on the geometry of the rectangle, the answer is option (1).
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Question: 2
AB > AF > BF; CD > DE > CE; and BF = \(6\sqrt{5}\) cm. Which is the closest pair of points among all the six given points?
AB > AF > BF; CD > DE > CE; and BF = \(6\sqrt{5}\) cm. Which is the closest pair of points among all the six given points?
Show Hint
Within each group of three points the closest pair is fixed by the given inequalities, so compare these candidates; none of the named pairs in the options can be the overall closest.
Updated On: Nov 4, 2026
- B, F
- None of these
- A, B
- C, D
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The Correct Option is B
Solution and Explanation
Step 1: Closest pair inside A, B, F:
AB > AF > BF means B, F is the closest pair among A, B, F, with \(BF = 6\sqrt{5} \approx 13.4\) cm.
Step 2: Closest pair inside C, D, E:
CD > DE > CE means the closest pair among C, D, E is C, E (not C, D).
Step 3: Cross pairs and the overall winner:
The points C, D, E lie in a \(5 \times 10\) rectangle, whose longest distance is \(\sqrt{125} \approx 11.2\) cm, so \(CE \le 11.2 < 13.4 = BF\). Pairs with one point from each rectangle are at least \(10\sqrt{3}\) cm apart, so they are not closer. The closest pair is therefore C, E (or possibly a cross pair), but in no case is it B, F; C, D; or A, B.
Step 4: Options:
(1) B, F is longer than CE. (4) C, D is longer than CE. (3) A, B is longer than BF. None of the listed pairs is the closest.
Final Answer:
The correct option is (2): None of these
AB > AF > BF means B, F is the closest pair among A, B, F, with \(BF = 6\sqrt{5} \approx 13.4\) cm.
Step 2: Closest pair inside C, D, E:
CD > DE > CE means the closest pair among C, D, E is C, E (not C, D).
Step 3: Cross pairs and the overall winner:
The points C, D, E lie in a \(5 \times 10\) rectangle, whose longest distance is \(\sqrt{125} \approx 11.2\) cm, so \(CE \le 11.2 < 13.4 = BF\). Pairs with one point from each rectangle are at least \(10\sqrt{3}\) cm apart, so they are not closer. The closest pair is therefore C, E (or possibly a cross pair), but in no case is it B, F; C, D; or A, B.
Step 4: Options:
(1) B, F is longer than CE. (4) C, D is longer than CE. (3) A, B is longer than BF. None of the listed pairs is the closest.
Final Answer:
The correct option is (2): None of these
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