Tangents from a point to a circle
From a point P outside a circle, draw the two lines that just touch it. Change the radius and how far P is from the centre, and find the length of each tangent and the angle between them.
Core concept
A tangent is perpendicular to the radius at the point where it touches, so the two tangents from an outside point are equal, each √(d² − r²)
Topic: Circle
A secant is a line that cuts a circle at two points. A tangent touches the circle at just one point, and at that point it is perpendicular to the radius. From a point P outside the circle, d from the centre O, two tangents can be drawn, PT₁ and PT₂. Each triangle OTP has a right angle: The amount of turn between two lines that meet, measured in degrees (°). All key words at T, so by Pythagoras PT² = OP² − OT² = d² − r², and PT = √(d² − r²). Both triangles have the same d and r, so the two tangents are equal. The angle between them is 2 × asin(r ÷ d): the farther P is, the narrower the angle.
What you will learn
- Tell a tangent from a secant.
- State that a tangent is perpendicular to the radius at the point of contact.
- Find the length of a tangent as √(d² − r²) and show that the two tangents from a point are equal.
Before you start
- Know Pythagoras' theorem and the parts of a circle.
Materials
- A compass, a ruler and a set square
Equipment
- None
Procedure
- Record a 6 cm circle with P 10 cm from the centre.
- Make the radius 5 cm and P 13 cm away, and record.
- Make the radius 3 cm, keep P 13 cm away, and record.
How this simulation is simplified
- P is always outside the circle, in the same plane.