Only one tangent
Turn a line about a point of a circle: just one position touches.
Many lines through P
Many lines can pass through a point P of a circle.
Turn a stick about P, one half turn. In most positions it cuts the circle again, at a second point.
How many of those positions touch the circle at P only?
What this lesson covers
The idea
At any point on a circle there is one and only one tangent.
Many lines through P
Many lines can pass through a point P of a circle.
Turn a stick about P, one half turn. In most positions it cuts the circle again, at a second point.
How many of those positions touch the circle at P only?
- Exactly one
- Two, one on each side
- A whole range of positions
Sweep the half turn
Turn the line about P with the knob H. The strip shows each position you have tried. Sweep the whole half turn, from one end to the other.
One and only one
At any point on a circle there is one and only one tangent: the line through the point perpendicular to the radius.
By Theorem 10.1 a tangent at P must be perpendicular to OP. Through P there is exactly one line perpendicular to OP. Every other line through P makes a different angle with OP, so it also meets the circle at a second point Q.
Notes
At any point on a circle there is one and only one tangent, the line perpendicular to the radius at that point.
Check yourself
How many tangents can be drawn at a given point of a circle?
Three different points are marked on a circle. How many tangents can be drawn at these three points in all?
Answer: 3
One tangent at each point, so 3 points give 3 tangents.
Why is the tangent at P the only line through P that does not meet the circle again?
A line through P on a circle makes an angle of 70° with the radius OP. How many common points do the line and the circle have?
Answer: 2
The angle is not 90°, so the line is not the tangent. It meets the circle at P and at a second point, so 2.
- None. There is always a tangent at a point of the circle: the line perpendicular to the radius there.
- Exactly one — correct. Yes! One and only one tangent at any point of a circle.
- Two. Two would need two different lines perpendicular to the same radius at P. There is only one such line.
- It is the only line through P perpendicular to the radius OP. — correct. Yes! Every other line through P makes a different angle with OP, and cuts the circle at a second point.
- It is the longest line through P.. Every line is endless. Length does not tell a tangent from a secant.
- It is the only line through the centre.. A line through the centre cuts the circle at two points. It is not a tangent.