Question

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.

Answer 1

The tangent drawn at the midpoint of an arc AMB is parallel to chord AB. This follows from equal opposite angles, ensuring AB is parallel to the tangent TMT'

1. Given:

- Arc AMB with midpoint M.

- Tangent TMT' to the circle.

- Joining lines AB and MB.

2. Claim:

-
`\( \text{Arc } AM = \text{Arc } MB \)`.

3. Proof of Claim:

- Since
`\( \text{Arc } AM = \text{Arc } MB \)`, then
`\( \angle MAB = \angle MBA \)` (Equal sides corresponding to equal angles) ....(i).

4. Assertion:

-
`\angle` AMT =
`\angle` MBA.

5. Proof of Assertion:

- Since TMT' is a tangent,
`\angle` AMT =
`\angle` MBA (Angles in alternate segments are equal).

- From equation (i),
`\angle` AMT =
`\angle` MAB.

- Therefore,
`\angle` AMT =
`\angle` MAB =
`\angle` MBA.

6. Conclusion:

- AB is parallel to TMT' (Alternate angles are equal).

Hence, the tangent drawn at the midpoint of an arc of a circle is parallel to the chord joining the endpoints of the arc.