Definition: A tangent to a circle is a line that touches the circle at exactly one point and is perpendicular to the radius drawn to that point.
What is a tangent line?
It grazes the circle at one point, the point of tangency, and does not enter the circle. Any other line through that point cuts the circle again, which makes it a secant.
Which facts matter?
- The radius to the point of tangency is perpendicular to the tangent.
- Two tangent segments from the same outside point are equal in length.
Example. A circle has center O and radius 7. A point P lies 25 units from O, and PT is tangent at T. Triangle OTP has a right angle at T, so PT2 = 252 – 72 = 625 – 49 = 576, and PT = 24. A second tangent from P to the circle also has length 24.
What mistakes occur?
Forgetting the right angle and treating OP as a leg instead of the hypotenuse. Another error is to confuse the tangent line with the tangent ratio in trigonometry, which is a different idea. To test whether a line is tangent, check that the right angle exists, using the converse of the Pythagorean theorem.
Where to learn it
- Parabolas from Focus and Directrix, and Tangents – lesson
- Circles: Angles, Arcs, Chords and Tangents – lesson