Put radius of circle any next keep centre at the point and cut arc and join the tangent

Understanding the Concept
To prove that the tangent drawn at the midpoint of an arc of a circle is parallel to the chord joining the endpoints of the arc, we start by visualizing a circle with center O and points A and B on the circumference. The arc AB is subtended by the chord AB.
Key Elements of the Proof
- Midpoint of the Arc: Let M be the midpoint of the arc AB. This means M is the point on the arc that divides it into two equal parts.
- Tangent at M: The tangent line at point M will be perpendicular to the radius OM, where O is the center of the circle.
- Chord AB: The chord AB connects points A and B on the circle.
Angles Involved
- Angle AOB: This is the central angle subtended by the chord AB.
- Angle OMA: The tangent at point M is perpendicular to the radius at that point, hence angle OMA = 90 degrees.
- Angle OMB: Similarly, angle OMB is also 90 degrees.
Relationship Between Angles
- Since M is the midpoint of the arc, the angles AOM and BOM are equal, each being half of angle AOB.
- The angle between the tangent at M (OM) and the chord AB is equal to the angle formed at O, between the radii OA and OB, thus establishing that the tangent line at M is indeed parallel to the chord AB.
Conclusion
Thus, through geometric relationships and properties of circles, it is proven that the tangent drawn at the midpoint of an arc is parallel to the chord that joins the endpoints of that arc. This fundamental property holds true for all circles and is a crucial concept in circle geometry.