How many tangents from a point?
A point inside a circle has no tangent, a point on it has one, a point outside has two.
Three kinds of points
A point P can be inside a circle, on the circle, or outside it. Imagine drawing a tangent through P each time.
Think about what a line through P does to the circle.
From which point can you draw two tangents to the circle?
What this lesson covers
The idea
No tangent to a circle passes through a point inside it, exactly one passes through a point on it, and exactly two pass through a point outside it.
Three kinds of points
A point P can be inside a circle, on the circle, or outside it. Imagine drawing a tangent through P each time.
Think about what a line through P does to the circle.
From which point can you draw two tangents to the circle?
- A point inside the circle
- A point on the circle
- A point outside the circle
Move the point
Drag the point P inside the circle, onto the circle and outside it. Watch how many tangents pass through P.
0, 1 or 2
Through a point inside a circle there is no tangent. Through a point on it there is exactly one. Through a point outside it there are exactly two.
Inside, every line through P cuts the circle at two points, so none of them is a tangent. On the circle there is one tangent, the line perpendicular to the radius. From outside, the tangents touch the circle at two points of contact, T₁ and T₂.
Notes
From a point inside a circle: no tangent. From a point on it: one. From a point outside it: two.
Check yourself
P is 3 cm from the centre of a circle of radius 5 cm. How many tangents can be drawn to the circle through P?
P is a point on a circle. How many tangents pass through P?
A point P is 13 cm from the centre of a circle of radius 5 cm. How many tangents can be drawn from P to the circle?
Answer: 2
13 > 5, so P is outside the circle. Exactly two tangents can be drawn from an outside point.
Why can no tangent be drawn through a point inside a circle?
- None — correct. Yes! P is inside the circle (3 < 5), and every line through P cuts the circle at two points.
- One. One tangent happens only when P is on the circle. Here P is inside.
- Two. Two tangents need a point outside the circle. Here P is closer to the centre than the radius.
- Exactly one — correct. Yes! The line perpendicular to the radius at P.
- Exactly two. Two tangents need a point outside the circle.
- None. P is on the circle, and the tangent at P passes through it.
- Every line through the point cuts the circle at two points. — correct. Yes! A tangent needs exactly one common point, and these lines have two.
- The point is too close to the centre to draw a line.. You can draw lines through any point. The problem is that they all cut the circle twice.
- Tangents only touch the circle at the centre.. A tangent touches the circle, not the centre.