Ab is the chord of circle diameter CD is perpendicular bisector of ab prove that 

Given:
- \( AB \) is a chord of the circle.
- \( CD \) is the diameter of the circle.
- \( CD \) is the perpendicular bisector of \( AB \).

### To Prove:
\( CD \) bisects \( AB \) perpendicularly.

### Proof:
1. **Given Information**:
   - Let \( O \) be the center of the circle, as \( CD \) is the diameter.
   - Since \( CD \) is a diameter, it passes through the center \( O \) of the circle.
   - The diameter of a circle always subtends a right angle to any point on the circle, so the angle formed by any point on the circle and a diameter is \( 90^\circ \).

2. **Right Angles Formed**:
   Since \( CD \) is perpendicular to \( AB \), this means that the angle \( \angle AOD = 90^\circ \) and \( \angle BOD = 90^\circ \).

3. **Perpendicular Bisector**:
   In a circle, a diameter that is perpendicular to a chord also bisects the chord. Thus, \( CD \) bisects \( AB \) at point \( M \), where \( M \) is the midpoint of chord \( AB \).
   
4. **Equality of Lengths**:
   As \( M \) is the midpoint of \( AB \), we have \( AM = MB \). This establishes that the diameter \( CD \) not only bisects the chord \( AB \) but does so perpendicularly.

5. **Conclusion**:
   Therefore, \( CD \), the diameter of the circle, is the perpendicular bisector of \( AB \).

### Q.E.D.

Would you like more details or have any questions?

### Related Questions:
1. What is the significance of the perpendicular bisector in a circle?
2. How do we prove the property of a diameter being perpendicular to a chord using coordinate geometry?
3. Why does the angle subtended by a diameter at the circumference always equal \( 90^\circ \)?
4. Can the perpendicular bisector property be extended to ellipses?
5. What is the relationship between the radius, the chord, and the perpendicular bisector?

### Tip:
In a circle, the perpendicular from the center to any chord always bisects the chord.

Ab is the chord of circle diameter CD is perpendicular bisector of ab prove that

Explore the geometric proof that the diameter of a circle is the perpendicular bisector of a chord. This detailed explanation covers key concepts in circle geometry, including the perpendicular bisector theorem and properties of a circle. Suitable for students in Grades 9-10.

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