Tangents and the number of tangents from a point
A tangent touches a circle at exactly one point; a secant cuts it at two. From a point outside the circle you can draw exactly two tangents.
A line and a circle: three cases
| Position of the line | Common points | Name |
|---|---|---|
| Does not meet the circle | 0 | Non-intersecting line |
| Cuts the circle | 2 | Secant |
| Just touches the circle | 1 | Tangent; the common point is the point of contact |
A tangent is a special secant
Draw a secant through two points A and B of a circle. Turn it slowly about A, so that B slides towards A. When B reaches A, the two points become one. The secant has become the tangent at A. So at every point of a circle there is exactly one tangent, and a circle has infinitely many tangents in all. The segment from an outside point P to the point of contact is called the length of the tangent from P.
Number of tangents from a point P
| Where P is | Number of tangents | Reason |
|---|---|---|
| Inside the circle | 0 | Every line through P cuts the circle at two points |
| On the circle | 1 | Only one line through P touches the circle there |
| Outside the circle | 2 | Two lines from P just touch the circle, one on each side |
How to draw tangents from an outside point (rough figure)
- Draw a circle and mark its centre O.
- Mark a point P outside the circle.
- With a ruler, draw two lines from P that just touch the circle, one on each side.
- Name the points of contact A and B.
- Join OA, OB and OP. Mark the right angles at A and B.
Counting tangents
A circle has centre O and radius 5 cm. Points P, Q and R are 3 cm, 5 cm and 8 cm from O. How many tangents can be drawn to the circle from each point?
- Compare each distance with the radius, 5 cm.
- OP = 3 cm < 5 cm, so P is inside the circle: 0 tangents.
- OQ = 5 cm = radius, so Q is on the circle: 1 tangent.
- OR = 8 cm > 5 cm, so R is outside the circle: 2 tangents.
P: 0 tangents, Q: 1 tangent, R: 2 tangents.
A bicycle tyre on a straight road touches the road at only one point at a time, so the road is a tangent. The rope of a well pulley also leaves the wheel along a tangent.
1 marks: “How many tangents can be drawn to a circle from a point inside it?” None (0). Every line through an inside point cuts the circle twice.
1 marks: “Fill in the blank: A tangent to a circle meets it at ____ point(s).” Exactly one.
2 marks: “Distinguish between a secant and a tangent to a circle, with a figure.” Secant: a line meeting the circle at two points. Tangent: a line meeting it at only one point, the point of contact. Draw one circle with both lines.
In 0, On 1, Out 2: the number of tangents from a point inside, on and outside a circle.