Perpendicular from the center of a circle to a chord bisects the chord

Equal chords subtend equal angles at the center of a circle	If Angles subtended by the chords at the center of circle are equal, then chords are also equal	Perpendicular from the center of a circle to a chord bisects the chord	Line drawn from the center of circle to bisect a chord, is perpendicular to the chord	Equal chords are equidistant from the center of circle
Chords equidistant from the center of circle are equal in length

Before you understand the property, the perpendicular from the center of a circle to a chord bisects the chord, you are advised to read: What are Chords of Circle ? What is center of Circle ? What is Perpendicular ? Observe the following diagram: In the above diagram, we have: A circle with center O XY is the chord of Circle M is the point of intersection of Line AB and Chord XY Now, you can also observe that: Line AB passes through the center O and Line AB is perpendicular to Chord XY So, as per the property of circle "The perpendicular from the center of a circle to a chord bisects the chord", so we get: Line AB bisects the chord XY Or we write it as: XM = MY Now, let's prove the property, The perpendicular from the center of a circle to a chord bisects the chord in the following way: Before you prove this property of circle, you are advised to read: What are Congruent Triangles ? What is RHS rule of Congruency ? What are the Corresponding Parts of Congruent Triangles ? What is the Radii of Circle ? Observe the following: In the above diagram, we have: A circle with center O XY is the chord of Circle Line AB passes through the center O M is the point of intersection of Line AB and Chord XY Line AB is perpendicular to Chord XY ∠ 1 = ∠ 2 = 90° each And we need to prove that: XM = MY Now, join point O & X and O & Y (as shown below) This would give us two triangles i.e. (as highlighted below): △ OMX = △ OMY ∠ 1 = ∠ 2 (90° each - Given) OX = OY (radii of circle are always equal) OM = OM (common side) Therefore, on applying RHS Rules of congruency, we get: ∆ OMX ≅ ∆ OMY Since, we know that corresponding parts of congruent triangles are equal, so we get: XM = MY Hence, this proves property of circle which says "The perpendicular from the center of a circle to a chord bisects the chord"