The normal

The idea

The line containing the radius through the point of contact is called the normal to the circle at that point.

The spoke and the road

A wheel rolls on the road. The road is the tangent. The spoke that points to the point of contact lies along a radius.

Extend that spoke into a long straight line.

Where does this line go after the point of contact, if it is extended through the wheel?

• Through the centre of the wheel
• Off to one side of the centre
• It depends on the wheel

Find the line through O

The orange line is the tangent at P. Drag the knob to turn the second line about P until it passes through the centre O.

The line along the radius

The line containing the radius through the point of contact is called the normal to the circle at that point.

The normal at P passes through the centre O. By Theorem 10.1 it is perpendicular to the tangent at P. In the toy the line turned green and a right angle appeared at the same moment.

Notes

The line containing the radius through the point of contact is the normal to the circle at that point. It passes through the centre and is perpendicular to the tangent.

Check yourself

Which line is the normal to a circle at a point P of the circle?

What is the angle between the normal and the tangent at the same point of a circle?

A circle has centre O and radius 7 cm. N is a point on the normal at P, outside the circle, with PN = 5 cm. How far is N from O (in cm)?

Answer: 12

O, P and N lie on the normal, with P between O and N. So ON = OP + PN = 7 + 5 = 12 cm.

Does the normal to a circle at a point pass through the centre?

• The tangent at P. The tangent touches the circle at P. The normal is a different line, the one along the radius.
• The line containing the radius through P — correct. Yes! That line is the normal at P.
• Any line through P. Most lines through P are secants. Only one of them goes through the centre and contains the radius.
• 0°, they are the same line. The normal goes through the centre. The tangent never does. They are two different lines.
• 60°. No such angle. The normal contains the radius, and the radius is perpendicular to the tangent.
• 90° — correct. Yes! The tangent is perpendicular to the radius (Theorem 10.1), and the normal contains the radius.
• Yes, always — correct. Yes! It contains the radius, so it goes through the centre.
• Only if the circle is small. The size of the circle does not matter.
• Never. The normal contains the radius through the point of contact, and a radius ends at the centre.