What is a tangent?

A tangent of a circle is a line that touches the circle at exactly one point. The blue circle on the left has a tangent on it, and the green circle on the right has a line going through the circle, but it is not tangent to the circle. The point T is called the point of tangency.

Find the tangent.

A circle is given below, with centre C, radius r. The radius r meets the circle at point P. Draw a tangent line at the point P, and add a point on the tangent. What is the angle between the tangent and the radius?

Using the Pythagorean Theorem with tangent lines.

We know that the angle between the radius and tangent is 90º, and the Pythagorean Theorem tells us about right angle triangles. Consider this problem. The line AB is tangent to the circle below, with A the point of tangency. The radius (AC) has length 4, and the line AB has length 7. What is the length of the line BC?