Tangent from an external point (construction)
Tangents from an External Point
Tangents from an External Point is a topic in Circles in the Common Core State Standards. It is aligned to Standard G-C.4, which requires students to construct a tangent line from a point outside a given circle.
The two tangent segments drawn to a circle from an external point are equal in length.
Theory
From a point outside a circle, exactly two tangent lines can be drawn. The key theorem:
Also, the line from the external point to the center bisects the angle between the two tangents.
The equal-tangent property:
How to use equal tangents
- Identify the two tangent segments from the external point.
- Set them equal and solve for the unknown.
- Use the angle bisector when the angle at the point is involved.
Two tangents from the same external point are congruent.
| \(\text{other tangent}\) | \(=\) | \(9\) |
Set the equal tangent lengths equal.
| \(3x-2\) | \(=\) | \(x+8\) |
| \(2x\) | \(=\) | \(10\) |
| \(x\) | \(=\) | \(5\) |
Equal tangents from a vertex, so add them.
| \(4+4\) | \(=\) | \(8\) |
It bisects the angle between the two tangents.
Common pitfalls
Frequently asked questions
Are the two tangents from an external point equal?
Yes. The two tangent segments from a single external point to a circle are congruent.
How many tangents can be drawn from an external point?
Exactly two.
What does the line to the center do?
It bisects the angle formed by the two tangents.
How do you solve for an unknown tangent length?
Set the two tangent expressions equal, since the tangents are congruent, and solve.