The tangent and the radius

The idea

Theorem 10.1: the tangent at any point of a circle is perpendicular to the radius through the point of contact.

A wheel on the road

A wheel rolls on a flat road. Its spokes lie along radii. The road is a tangent to the wheel.

Look at the spoke that points straight down to the road, at the point of contact.

What angle does that spoke make with the road?

• A right angle, 90°
• A sharp angle, less than 90°
• It is different for each wheel

Turn the line to touch

P is a point on the circle. Drag the knob H to turn the line about P. When the line touches the circle at one point only, read the angle between the line and the radius OP.

Radius and tangent make 90°

Theorem 10.1. The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Why? Take any other point Q on the tangent. Q must be outside the circle, because if Q were inside, the line would be a secant. So OQ is longer than the radius OP. This holds for every point of the line except P, so OP is the shortest distance from O to the line. The shortest distance is the perpendicular, so OP ⟂ XY.

Notes

Theorem 10.1. The tangent at any point of a circle is perpendicular to the radius through the point of contact. OP is the shortest distance from the centre O to the tangent.

Check yourself

A tangent touches a circle at P and O is the centre. What is the angle between the tangent and the radius OP?

Q is a point on the tangent at P, other than P. How do OQ and the radius OP compare?

The tangent at P touches a circle of radius 5 cm with centre O. Q is a point on the tangent with OQ = 13 cm. How long is PQ (in cm)?

Answer: 12

Angle OPQ = 90°, so PQ² = OQ² − OP² = 13² − 5² = 169 − 25 = 144, and PQ = 12 cm.

PQ is a tangent at P to a circle with centre O. Angle POQ = 50°. Find angle OQP (in degrees).

Answer: 40

Angle OPQ = 90°. So angle OQP = 180° − 90° − 50° = 40°.

• 45°. The angle is not half a right angle. Think of the spoke of a wheel and the road: they meet straight.
• 90° — correct. Yes! The tangent is perpendicular to the radius through the point of contact (Theorem 10.1).
• It depends on the size of the circle.. The size of the circle makes no difference. The tangent is perpendicular to the radius in every circle.
• OQ is shorter than OP.. If OQ were shorter than the radius, Q would be inside the circle, and the line would be a secant, not a tangent.
• OQ is equal to OP.. If OQ equalled the radius, Q would be on the circle, and the line would have a second common point.
• OQ is longer than OP. — correct. Yes! Every point of the tangent other than P is outside the circle, so OQ > OP.