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What is the Locus of Points Which Are Equidistant from the Given Non-collinear Point A, B and C? Justify Your Answer. - Mathematics

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Question

What is the locus of points which are equidistant from the given non-collinear point A, B and C? Justify your answer.

Sum

Solution

Let A, B, C be three distinct points on a line1. Any point equidistant from A and B lies on the perpendicular bisector of Ab. So, points equidistant from A and B lies on line m.
Similarly, points equidistant from B and C lies on line n which is the perpendicular bisector of BC.
Thus, any point equidistant from A, B and C must be common to both the lines m and n.

But m ⊥ AB and n ⊥ BC.
∴ m ⊥ AC and n ⊥ AC
⇒ m || n
So, no points is common to both m and n.
Hence, the required locus is the null set Φ.

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Chapter 14: Loci (Locus and its Constructions) - Figure Based Questions

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ICSE Mathematics [English] Class 10
Chapter 14 Loci (Locus and its Constructions)
Figure Based Questions | Q 13

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