Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Step-by-Step Solution
Key Concept: A tangent to a circle is perpendicular to the radius at the point of contact. Radii drawn to the endpoints of a diameter lie on the same straight line; therefore the two tangents are each perpendicular to the same line, making them parallel.
1. Let $O$ be the centre of the circle and $AB$ be a diameter.\
2. Draw the tangents at the end points $A$ and $B$; let them intersect the extended line $AB$ at points $P$ and $Q$ respectively.\
3. By the definition of a tangent, the radius $OA$ is perpendicular to the tangent at $A$; hence \[ OA \perp AP. \] Similarly, the radius $OB$ is perpendicular to the tangent at $B$; thus \[ OB \perp BQ. \]\
4. Since $AB$ is a diameter, the points $A$, $O$, and $B$ are collinear, i.e., \[ OA \text{ and } OB \text{ lie on the same straight line}. \]\
5. Both $AP$ and $BQ$ are perpendicular to the same straight line $AB$.\
6. If two lines are each perpendicular to a third line, they are parallel to each other. Therefore, \[ AP \parallel BQ. \]\
7. Hence, the tangents drawn at the ends of a diameter of a circle are parallel.
Correct Answer: The tangents at the ends of a diameter are parallel because each is perpendicular to the same diameter.