Question 31 of 50intermediate🔧 ApplyLong Answer5 marks

Prove that the perpendicular bisector of a chord passes through the centre of the circle.

Correct Answer

Let ABAB be a chord of a circle with centre CC. Let MM be the midpoint of the chord ABAB. We need to prove that the perpendicular bisector of ABAB passes through the centre CC.

By Theorem 4, the line joining the centre of a circle and the midpoint of a chord is perpendicular to the chord. Therefore, the line segment CMCM joining the centre CC and the midpoint MM is perpendicular to the chord ABAB.

Since MM is the midpoint of ABAB and CMCM is perpendicular to ABAB, the line CMCM bisects ABAB and is perpendicular to it. Thus, the line CMCM is exactly the perpendicular bisector of the chord ABAB.

Through a given point MM on the line ABAB, there is exactly one line perpendicular to ABAB. This means the perpendicular bisector of ABAB is unique, and it must coincide with the line CMCM.

Since the centre CC lies on the line CMCM, and CMCM is the perpendicular bisector of ABAB, it directly follows that the perpendicular bisector of the chord passes through the centre of the circle. Hence proved.

Exercise: END-OF-CHAPTER EXERCISES | Q: 5 | (Page 23)
For More Understanding

Explanation

The proof relies on Theorem 4 from the textbook, which establishes that the line joining the centre to the midpoint of a chord is perpendicular to the chord. By combining this with the fact that a line passing through the midpoint and perpendicular to the segment is its perpendicular bisector, and the uniqueness of such a perpendicular line, we conclude that the centre must lie on the perpendicular bisector.

Solution Steps

  1. Step 1: Let ABAB be a chord of a circle with centre CC, and let MM be the midpoint of ABAB.

  2. Step 2: By Theorem 4, the line joining the centre CC and the midpoint MM is perpendicular to the chord ABAB. Thus, CMCM is perpendicular to ABAB.

  3. Step 3: Since MM is the midpoint of ABAB and CMCM is perpendicular to ABAB, the line CMCM is the perpendicular bisector of ABAB.

  4. Step 4: Because there is only one perpendicular bisector for a given line segment, and the centre CC lies on this line CMCM, the perpendicular bisector of the chord must pass through the centre of the circle.

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