Tangents to a Circle
A tangent is a line that touches a circle at exactly one point.
That point is called the point of tangency.
The Radius is Perpendicular
A very important rule is that the radius to the point of tangency is always perpendicular (at right angles) to the tangent line.
This means the angle between them is always 90°.
Try it: Draw a circle, and its radius, and then use a protractor to draw a line at 90°. That line will be a tangent!
Two Tangents from one Point
When we draw two tangent lines from the same outside point to a circle:
- The distance from the point to the circle is the same for both (a)
- The angles they make with the center line are also the same (θ)
Common Tangents
When we have two circles, we can draw lines that are tangent to both of them at the same time. These are called Common Tangents.
External (or Direct) Common Tangents
These stay on "the outside" and don't cross the space between the circles.
Internal (or Transverse) Common Tangents
These cross through the space between the two circles.
Other Cases
We can also have these cases:
Circles just touch, outside
Circles intersect
Circles just touch, inside
Circle inside circle
Summary
- A tangent touches a circle at just one point
- It is always at 90° to the radius
- Two tangents from the same point have the same length
- Common tangents can be external (stay outside) or internal (cross over)
Now you know the first secrets of the tangent!