Proving that the right bisector of a chord of a circle bisects the corresponding arc on the circle

To prove that the right bisector of a chord of a circle bisects the corresponding arc on the circle, we will use the following steps:

Step 1: Define the right bisector and chord
- The right bisector is a line that cuts another line segment into two equal parts at a right angle.
- The chord is a line segment that connects two points on the circumference of a circle.

Step 2: Draw a diagram
- Draw a circle and mark its center as O.
- Draw a chord AB on the circle.
- Construct the perpendicular bisector of AB, which is a line that passes through the midpoint of AB and forms a right angle with AB.
- Let the midpoint of AB be M and the perpendicular bisector intersect the circle at point P.

Step 3: Prove that PM is a radius
- Since M is the midpoint of AB, AM = MB.
- As the perpendicular bisector of a line segment cuts it into two equal parts, PM = PB.
- Therefore, PM = PB = radius of the circle.

Step 4: Prove that the perpendicular bisector bisects the corresponding arc
- To prove that the perpendicular bisector of AB bisects the corresponding arc, we need to show that the lengths of arc AP and arc BP are equal.
- From the previous step, we know that PM = PB.
- Since O is the center of the circle, OP is also a radius. Therefore, OP = OA = OB.
- Hence, arc AP = arc OP and arc BP = arc OP.
- Therefore, arc AP = arc BP, which means the perpendicular bisector of AB bisects the corresponding arc on the circle.

Step 5: Conclusion
- The right bisector of a chord of a circle bisects the corresponding arc on the circle.
- This is because the perpendicular bisector cuts the chord into two equal parts and intersects the circle at points equidistant from the center, making the lengths of the corresponding arcs equal.

Note: The proof assumes that the chord is not a diameter of the circle. If the chord is a diameter, the right bisector coincides with the diameter and still bisects the corresponding arc.