2.Let a circle with center O and a point P on the circle and make a line perpendicular to radius OP. Take any point Q on the line. How many tangents can be drawn to the circle at a point on the circle?


In this problem, you're asked to consider a circle with center \( O \), a point \( P \) on the circle, and a line perpendicular to the radius \( OP \). Then, you are to explore how many tangents can be drawn from any point \( Q \) on the line perpendicular to \( OP \) to the circle.

### Step-by-Step Breakdown:

1. **Circle and Tangents**:
   - The general rule for tangents to a circle from a point outside the circle is that exactly two tangents can be drawn from any external point to a circle.
   
2. **Given Configuration**:
   - We are given that \( P \) is a point on the circle, and the line perpendicular to \( OP \) is constructed.
   - The perpendicular line will not pass through the center \( O \), so any point \( Q \) on this line (excluding \( P \)) will lie outside the circle.

3. **Tangents from Point \( Q \)**:
   - Since \( Q \) is external to the circle (as it lies on the perpendicular line), exactly **two tangents** can be drawn from \( Q \) to the circle, regardless of where \( Q \) is chosen on the line, as long as it is not at point \( P \).

Thus, from any point \( Q \) on the perpendicular line (except at \( P \)), **two tangents** can be drawn to the circle.

### Final Answer:
The number of tangents that can be drawn from point \( Q \) on the perpendicular line to the circle is **2**.

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Would you like more details on how tangents to circles work or any specific part of the reasoning?

### Follow-up questions:
1. What is the geometric condition for a line to be tangent to a circle?
2. How can we prove that exactly two tangents can be drawn from an external point to a circle?
3. Can a tangent line pass through the center of the circle?
4. How would the situation change if \( Q \) were inside the circle instead of on the perpendicular line?
5. How would you construct the tangents from an external point using geometric tools?

### Tip:
Remember that a tangent to a circle always touches the circle at exactly one point and is perpendicular to the radius at the point of contact.

Let a circle with center O and a point P on the circle and make a line perpendicular to radius OP. Take any point Q on the line. How many tangents can be drawn to the circle at a point on the circle?

Explore the geometry of circles and tangents with this problem. Learn how to determine the number of tangents that can be drawn from a point on a line perpendicular to the radius of a circle. A detailed breakdown of the tangent-secant theorem and the properties of tangents is provided. Suitable for students in Grades 9-12.

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