Radius is perpendicular to the tangent of the circle

Proof:

Let’s consider a circle with center and a tangent that touches the circle at point . We need to prove that the radius is perpendicular to the tangent at the point .

1. Definition of a Tangent: A tangent to a circle is a straight line that touches the circle at exactly one point. By definition, the tangent does not cross the circle at that point.

2. Assume Two Points on the Circle: Suppose there’s a point on the tangent line other than . Since is the only point of contact between the circle and the tangent, point must lie outside the circle.

3. Compare Distances from the Center: The distance from the center to any point on the circle is the radius. Therefore, the distance is the radius of the circle. The distance from the center to any other point on the tangent is greater than , because the line from to passes outside the circle.

4. Conclusion About Perpendicularity: The point is the only point where the distance from to the tangent line is minimized (equal to the radius). By the property of tangents and circles, this shortest distance between a point and a line is always along the perpendicular. Therefore, the line (the radius) must be perpendicular to the tangent line at .

Hence, the radius of a circle is perpendicular to the tangent at the point of contact.