Where do they touch?

The idea

The common point of a tangent and the circle is the point of contact, and the tangent is said to touch the circle at that point.

A wheel on the road

A bicycle wheel rolls along a straight road. Think of the wheel as a circle and the road as a straight line.

The wheel and the road touch.

At one moment, how many points do the circle and the line have in common?

• Exactly one
• Exactly two
• A whole stretch of points

Tap the point of contact

Tap the spot where the line meets the circle. If you tap the wrong place, the toy tells you what is missing.

On the line and on the circle

The common point of a tangent and the circle is called the point of contact, and the tangent is said to touch the circle at that point.

The point of contact is on the line and on the circle. That is why a tap on the line alone, or on the circle alone, was not enough. A wheel moves along a line that is a tangent to it, and the wheel touches the road at the point of contact.

Notes

The point of contact is the common point of a tangent and the circle. The tangent touches the circle at that point.

Check yourself

A tangent touches a circle at a point P. What is P called?

Which is true about the point of contact of a tangent?

A wheel rolls along the road. Where is its point of contact with the road?

How many points of contact can one tangent have with a circle?

• The point of contact — correct. Yes! It is the common point of the tangent and the circle.
• The centre. The centre is inside the circle. The tangent does not even reach it.
• The secant point. There is no such name. A secant has two common points with the circle, not one.
• It is on the tangent but inside the circle.. A point inside the circle is not on the circle. If the tangent went inside, it would cut the circle.
• It is on the tangent and on the circle. — correct. Yes! It is the one point that both share.
• It is the centre of the circle.. The centre is not on the circle, so it cannot be a common point of the circle and a line that only touches it.
• At the hub, the centre of the wheel. The hub is far above the road. It is not on the road.
• At the top of the wheel. The top is the farthest point from the road.
• At the point where the wheel touches the road — correct. Yes! That is the one point that is on the wheel and on the road.
• Two. A line with two common points is a secant, not a tangent.
• Exactly one — correct. Yes! A tangent meets the circle at one point only.
• None. A tangent always touches the circle, so it has one point of contact.