Construct a Tangent Line to a Point on a Circle

To construct a tangent to a given circle at a specified point on its circumference, follow these steps:

  1. Identify Key Elements
    • Let the given circle have center O.
    • Let P be the point on the circle where the tangent will be drawn.
  2. Draw the Radius to the Point of Tangency
    • Use a straightedge to draw a line from the center O through the given point P on the circle.
    • This radius OP will serve as a reference because a tangent to a circle is always perpendicular to the radius at the point of tangency.
  3. Construct a Perpendicular Line at P
    • Place the compass at P and draw a small arc that intersects the radius OP at two points, one on each side of P.
    • Keeping the same compass width, place the compass on each of these intersection points and draw two more arcs that cross each other on the opposite side of P from O.
    • Use the straightedge to draw a line through P and the intersection of these arcs.
  4. Result: The Tangent Line
    • The line constructed through P is perpendicular to OP, making it the tangent to the circle at P.

Animation:

Why This Works

  • By definition, a tangent to a circle is perpendicular to the radius at the point of tangency.
  • The perpendicular construction ensures an exact 90-degree angle at P, guaranteeing that the line is a true tangent.