How can we prove that the perpendicular line at the point of contact of a tangent to a circle passes through the center of the circle?

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To prove that the perpendicular drawn at the point of contact of a tangent to a circle passes through the center, we can follow a step-by-step geometric proof using basic properties of a circle.

Definitions:

  • Let O be the center of the circle.
  • Let T be the point where the tangent touches the circle.
  • Let r be the radius of the circle, i.e., the distance from O to T.

Steps of the Proof:

  1. Draw the radius: Connect the center O to the point of contact T. This line segment, OT, represents the radius of the circle.
  2. Draw the tangent: Imagine or draw a line that touches the circle at the point T without crossing it. This line is called the tangent to the circle.
  3. Angle Analysis: By the property of tangents to circles, we know that the radius at the point of contact is always perpendicular to the tangent line. Therefore, the angle ∠OTC (where C is any point along the tangent) is 90 degrees.
  4. Perpendicular Line: Now draw a perpendicular line from point T to line OT. This line will certainly pass through point O, as we established earlier that OT is perpendicular to TC.
  5. Conclusion: Since OT is the radius and is perpendicular to the tangent line at the point T, this means that the line from O to T (OT) must colloquially extend vertically from the tangent, proving our assertion.

Thus, we can conclude that the perpendicular at the point of contact of the tangent to a circle indeed passes through the center of the circle.

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