Does the line touch the circle?

The idea

A line and a circle in a plane have no common point (non-intersecting line), two common points (secant) or exactly one common point (tangent); no other position is possible.

A ruler and a coin

Put a coin on the table and slide a ruler across it, from one side to the other.

At first the ruler is away from the coin. Later it cuts through the coin. Somewhere in between, the ruler only touches the edge.

How many points can a straight line and a circle have in common?

• Only none or two
• None, one or two
• Any number, even five

Slide the line

Drag the orange knob to slide the line across the circle. Watch how many points the line and the circle share.

Three positions, no more

A line and a circle in a plane have no common point (a non-intersecting line), two common points (a secant) or exactly one common point (a tangent). No other position is possible.

A secant cuts the circle at two points A and B. A tangent touches the circle at only one point. The rope on both sides of a pulley is like a tangent to the pulley wheel.

Notes

A line and a circle have no common point (non-intersecting line), two (secant) or exactly one (tangent). There is no other position.

Check yourself

A line has two common points A and B with a circle. What is the line called?

A line and a circle have exactly one common point. What is the line called?

Can a straight line cut a circle at three different points?

The rope on each side of a pulley just touches the wheel at one point. The rope is like a…

• A tangent. A tangent has only one common point with the circle.
• A secant — correct. Yes! A line that cuts the circle at two points is a secant.
• A non-intersecting line. A non-intersecting line has no common point with the circle.
• A secant. A secant cuts the circle at two points.
• A non-intersecting line. That line has no common point at all.
• A tangent — correct. Yes! A tangent touches the circle at exactly one point.
• Yes, if the line is long enough.. Making the line longer does not help: a straight line meets a circle at no more than two points.
• No. A line and a circle have none, one or two common points. — correct. Yes! There is no other position of a line with respect to a circle.
• Yes, but only if it passes through the centre.. A line through the centre still meets the circle at exactly two points.
• tangent — correct. Yes! It touches the circle at one point only.
• secant. A secant cuts through the circle at two points. The rope does not cut through the wheel.