47. Secants & Tangents · Two lines, one circle, infinite possibilities

Try to break it

Grab the grip S and slide it up or down. Watch the secant line move! As it gets closer to the edge, the two intersection points M and N drift together. When they finally meet at one point, the secant transforms into a tangent. Cool, right?

How you build it

Draw a secant and a tangent to a circle.

• Draw a circle — click once for the centre O, then click again to set the radius.
• Mark a point A on the circle.
• Mark a second point B on the circle, well away from A.
• Draw a straight line through A and B. Because it passes through two points of the circle, it cuts the circle at two points — this is a secant.
• Mark a point T on the circle — this will be the point of contact for the tangent.
• Draw a straight line through T: click T, then click a second point so the line just touches the circle at T without cutting across it — this is a tangent.

The proof, step by step

Prove that a tangent meets the circle at one point and a secant at two.

• A secant is a straight line that intersects a circle at two distinct points.
• A tangent is a straight line that touches a circle at exactly one point, known as the point of contact.
• When a secant moves closer to the circle's edge until its two intersection points merge into one, it becomes a tangent.

Worked example

In the given figure, line l touches the circle at point P. Line m intersects the circle at points A and B. Which of the following statements is correct?

By definition, a line touching a circle at exactly one point is a tangent, while a line intersecting it at two points is a secant. Thus, l is a tangent and m is a secant.

• Line l is a secant and line m is a tangent.
• Line l is a tangent and line m is a secant. — correct
• Both lines are tangents.
• Both lines are secants.