Prove that a tangent line is always perpendicular to the point of contact.

To prove that the tangent line is perpendicular to the radius at the point of contact on a circle, let's go through a geometric proof.

### Setup and Understanding
Consider a circle with center \( O \) and a point \( P \) on the circumference. Let \( T \) be the tangent line to the circle at point \( P \).

### Statement to Prove
We aim to prove that the tangent line \( T \) is perpendicular to the radius \( OP \) at the point of contact \( P \).

### Proof
1. **Definition of Tangent Line**: A tangent to a circle at any point \( P \) is a line that touches the circle at exactly one point, meaning it does not cross the circle. Therefore, the tangent line \( T \) at \( P \) does not intersect the circle at any point other than \( P \).

2. **Distance Minimization Property**: The shortest distance between a point and a line is along the line perpendicular to it. Since \( OP \) is the radius of the circle, it is the shortest line segment from the center \( O \) to any point on the circumference, including the point \( P \) where the tangent touches the circle.

3. **Unique Intersection Property of Tangents**: Suppose we have any other line passing through \( P \) at an angle to \( OP \). This line would intersect the circle at another point because the angle allows it to enter the circle's interior, contradicting the definition of a tangent.

4. **Right Angle Condition**: The only way for \( T \) to be tangent (i.e., not cross into the circle) is if it meets \( OP \) at a right angle, making \( T \) perpendicular to \( OP \).

5. **Conclusion**: Thus, the tangent line \( T \) at point \( P \) must be perpendicular to the radius \( OP \) to satisfy the properties of a tangent line on a circle.

### Summary
This proof shows that a tangent line to a circle at any point of contact is perpendicular to the radius at that point, which is a fundamental property of circles.

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Do you have any questions on the proof? Here are some additional questions you may find interesting:

1. Why is the radius the shortest distance from the center to the tangent line?
2. How can we apply this perpendicular property in finding the slope of a tangent line?
3. What changes in this proof if we consider other types of curves, like an ellipse?
4. How does this concept apply to 3D objects, like spheres?
5. Can we use calculus to find the slope of the tangent and verify perpendicularity?

**Tip:** Remember, for any point on a circle, the tangent line’s slope is the negative reciprocal of the radius’ slope, confirming perpendicularity algebraically!

Explore the proof that a tangent line to a circle is always perpendicular to the radius at the point of contact. This fundamental geometric concept is essential for understanding circle properties and tangent behavior, suitable for high school geometry students in Grades 10-12.

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