C
CraftExam
high importancemedium8 min

Three Points Determine a Circle

One and only one circle can pass through three non-collinear points. Its centre is found at the intersection of perpendicular bisectors of two joining segments.

Practice This Concept

Learn the concept

Student-friendly explanation

Three points that do not lie on one straight line form a triangle. The perpendicular bisectors of its sides meet at one point, which is equidistant from all three points. That point becomes the centre of the required circle.

How to write this in exams

  1. 1

    Start with the exact idea

    One and only one circle can pass through three non-collinear points. Its centre is found at the intersection of perpendicular bisectors of two joining segments.

  2. 2

    Then show how to use it

    Join the three points to form two segments. Construct perpendicular bisectors of two segments. Mark their intersection as centre. Draw circle using distance from centre to any one point.

  3. 3

    Add one concrete example

    For non-collinear points A, B and C, draw perpendicular bisectors of AB and BC. If they meet at O, then OA = OB = OC, so a circle with centre O passes through A, B and C.

  4. 4

    Avoid this incomplete answer

    Choosing the midpoint of only one side as centre is wrong because it may be equidistant from two points, not necessarily from the third.

Definition

One and only one circle can pass through three non-collinear points. Its centre is found at the intersection of perpendicular bisectors of two joining segments.

Example

For non-collinear points A, B and C, draw perpendicular bisectors of AB and BC. If they meet at O, then OA = OB = OC, so a circle with centre O passes through A, B and C.

Rule to remember

Unique circle condition: three points must be non-collinear. Centre: intersection of perpendicular bisectors of two sides.

Memory hook

Three non-straight points fix one circle.

Examples and method

Worked example

Given three non-collinear points P, Q, R. Construct perpendicular bisectors of PQ and QR. Their intersection O gives OP = OQ = OR, so draw the circle with centre O and radius OP.

Method to apply

Join the three points to form two segments. Construct perpendicular bisectors of two segments. Mark their intersection as centre. Draw circle using distance from centre to any one point.

Diagram support

Draw triangle ABC, perpendicular bisectors of AB and BC meeting at O, then draw circle through A, B and C.

How CBSE asks it

Exam questions may ask for construction of circumcircle of a triangle or proof that a circle through three non-collinear points is unique.

Avoid common mistakes

Common confusion

Students forget the condition non-collinear. Three points on a straight line cannot lie on one ordinary circle.

Common wrong answer

Choosing the midpoint of only one side as centre is wrong because it may be equidistant from two points, not necessarily from the third.

Exam tip

In construction or proof, mention that the centre is equidistant from all three points because it lies on perpendicular bisectors.

Quick check

Why can three collinear points not determine a circle?

Three collinear points cannot determine a circle because a circle would need one centre equidistant from all three, but the perpendicular bisectors do not meet in a suitable single centre for a straight-line set of three points.

Answer writing and exam use

1-mark answer

One and only one circle can pass through three non-collinear points. Its centre is found at the intersection of perpendicular bisectors of two joining segments.

2-mark answer

One and only one circle can pass through three non-collinear points. Its centre is found at the intersection of perpendicular bisectors of two joining segments. Unique circle condition: three points must be non-collinear. Centre: intersection of perpendicular bisectors of two sides. For non-collinear points A, B and C, draw perpendicular bisectors of AB and BC. If they meet at O, then OA = OB = OC, so a circle with centre O passes through A, B and C.

3-mark answer

Three points that do not lie on one straight line form a triangle. The perpendicular bisectors of its sides meet at one point, which is equidistant from all three points. That point becomes the centre of the required circle. Unique circle condition: three points must be non-collinear. Centre: intersection of perpendicular bisectors of two sides. Given three non-collinear points P, Q, R. Construct perpendicular bisectors of PQ and QR. Their intersection O gives OP = OQ = OR, so draw the circle with centre O and radius OP. Exam questions may ask for construction of circumcircle of a triangle or proof that a circle through three non-collinear points is unique. Choosing the midpoint of only one side as centre is wrong because it may be equidistant from two points, not necessarily from the third.
MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept