The Tangent at any Point of a Circle is Perpendicular to the Radius through the Point of Contact

The statement mentioned above is a fundamental property of circles that holds true for all circles. It states that if a line is drawn tangent to a circle at any point, then the line is perpendicular to the radius through the point of contact.

Explanation:

To understand this property, let's consider a circle with its center at point O and a tangent line drawn to the circle at point A.

1. Radius and Tangent Line
- A radius is a line segment that connects the center of the circle to any point on the circumference.
- In this case, the radius is line OA.
- The tangent line is a line that touches the circle at only one point, in this case, point A.

2. Hypothesis:
- Let's assume that the tangent line is not perpendicular to the radius through the point of contact.
- In other words, let's assume the tangent line is not at a right angle to the radius OA.

3. Construction:
- Now, let's draw a line segment AB that is perpendicular to the radius OA at point A.
- Since AB is perpendicular to OA, angle OAB is a right angle.

4. Proof by contradiction:
- If the tangent line is not perpendicular to the radius, then the line AB and the tangent line should meet at some point B.
- However, this contradicts the definition of a tangent line, as a tangent line should only touch the circle at one point.
- Therefore, our assumption that the tangent line is not perpendicular to the radius is false.

Conclusion:
- Since assuming the opposite leads to a contradiction, we can conclude that the tangent line drawn to a circle at any point is perpendicular to the radius through the point of contact.
- This property holds true for all circles, irrespective of their size or position.

Importance:
- This property is crucial in various mathematical applications and problem-solving involving circles.
- It helps in determining angles, finding the lengths of line segments, and solving geometric problems related to circles.
- Understanding this property is essential for a solid foundation in geometry and trigonometry.