Can you explain why the converse to Theorem 4 is true, i.e., why does the perpendicular from the centre Class 9

Can you explain why the converse to Theorem 4 is true, i.e., why does the perpendicular from the centre Class 9

Question 1.

Can you explain why the converse to Theorem 4 is true, i.e., why does the perpendicular from the centre of a circle to a chord of the circle bisect the chord? (Hint: Use fig. 5.12. You are told that ∠CMA = ∠CMB = 90°. You need to show that AM = BM.) Class 9

Answer:

The perpendicular from the centre of a circle to a chord of the circle bisects the chord.

Given: AB is a chord of a circle.

CM is perpendicular to chord AB.

To show: M is the midpoint of AB.

Proof:

In ∆CMA and ∆CMB,

CA = CB ...[Radii of the same circle]

CM = CM ... [Common side]

∠CMA = ∠CMB = 90° ... [CM is perpendicular to AB]

∴ ∆CMA ≅ ∆CMB ...[By RHS congruence]

∴ AM = MB ... [Corresponding parts of congruent triangles]

∴ M is the midpoint of AB.