Understanding the Circle and Diameter
A circle is defined as the set of all points equidistant from a fixed point called the center. A diameter is a straight line segment that passes through the center and connects two points on the circle.
Defining Tangents
A tangent to a circle is a straight line that touches the circle at exactly one point. The key property of a tangent is that it is perpendicular to the radius drawn to the point of tangency.
Identifying the Points
- Let O be the center of the circle.
- Let A and B be the endpoints of the diameter.
- Let P and Q be the points where the tangents touch the circle at points A and B, respectively.
Using the Properties of Tangents
1. Tangents and Radii:
- The radius at point A (OA) is perpendicular to the tangent at point A (line AP).
- The radius at point B (OB) is perpendicular to the tangent at point B (line BQ).
2. Angles Formed:
- Since OA ⊥ AP, angle OAP = 90 degrees.
- Since OB ⊥ BQ, angle OBQ = 90 degrees.
Applying Parallel Line Theorem
- The angles OAP and OBQ are both 90 degrees.
- Since both tangents AP and BQ form equal angles with the radius, they are parallel to each other (as they are both perpendicular to the same line, the diameter AB).
Conclusion
Thus, it is proven that the tangents drawn at the endpoints of a diameter of a circle are indeed parallel. This geometric property is essential in understanding relationships within circles and their tangents.