Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.
Tangent at mid-point of arc is parallel to chord.
Step 1: Draw a circle with centre O. Mark an arc AB on the circle and let M be the mid-point of arc AB.
Step 2: Join AM and BM. Also join the chord AB.
Step 3: At point M, draw a tangent line (let’s call it t).
Step 4: By the Alternate Segment Theorem, the angle made by the tangent at a point on the circle with a chord through that point is equal to the angle made in the opposite arc.
Step 5: Apply this theorem for tangent t at M. The angle between tangent t and chord AM is equal to angle ∠ABM.
Step 6: Similarly, the angle between tangent t and chord BM is equal to angle ∠BAM.
Step 7: Since M is the mid-point of arc AB, angles ∠ABM and ∠BAM are equal.
Step 8: This shows that tangent t makes equal angles with lines AM and BM. Therefore, tangent t is parallel to chord AB.
Final Conclusion: The tangent at the mid-point of an arc is parallel to the chord joining the ends of the arc.