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Tangent to a circle at a point
Geometry construction using a compass and straightedge

This shows how to construct the tangent to a circle at a given point on the circle with compass and straightedge or ruler. It works by using the fact that a tangent to a circle is perpendicular to the radius at the point of contact. It first creates a radius of the circle, then constructs a perpendicular to the radius at the given point.

Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

Proof

The image below is the final drawing above.

  Argument Reason
1 Line segment OR is a radius of the circle O. It is a line from the center to the given point P on the circle.
2 SP is perpendicular to OR By construction, SP is the perpendicular to OR at P. See Constructing a perpendicular to a line at a point for method and proof.
3 SP is the tangent to O at the point P The tangent line is at right angles to the radius at the point of contact. See Tangent line definition.

  - Q.E.D

Try it yourself

Click here for a printable tangents problem worksheet with some problems to try. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
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Other constructions pages on this site

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Non-Euclidean constructions