Q1. [2] § 7.1 Introduction § 7.2 Distance Formula § 7.4 Summary
Using distance formula, prove that the points $A(2, 3)$, $B(-7, 0)$ and $C(-1, 2)$ are collinear.
Previously asked in CBSE board exam
2026 30/5/1 Q22(b) (OR-2)
Generated by claude-sonnet-4-6 · 2026-06-15 10:32 · grounding rag
Model Answer
Using the distance formula:
$$AB = \sqrt{(-7-2)^2 + (0-3)^2} = \sqrt{81+9} = \sqrt{90} = 3\sqrt{10}$$
$$BC = \sqrt{(-1-(-7))^2 + (2-0)^2} = \sqrt{36+4} = \sqrt{40} = 2\sqrt{10}$$
$$AC = \sqrt{(-1-2)^2 + (2-3)^2} = \sqrt{9+1} = \sqrt{10}$$
Since $BC + AC = 2\sqrt{10} + \sqrt{10} = 3\sqrt{10} = AB$, the points A, B and C are collinear. Proved.
Source: Chapter 7, Section 7.2 (Distance Formula)
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Explanation
- Collinearity condition: Three points are collinear if the sum of any two distances equals the third (i.e., one point lies between the other two).
- Always compute all three distances and check which two add up to the third.
- Here C lies between B and A, so AC + BC = AB.
- Show the arithmetic clearly — examiners award marks for each correct distance and the final conclusion.
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