Tangent lines (radius perpendicular at point of tangency)
Tangent Lines to a Circle
Tangent Lines to a Circle is a topic in Circles in the California Common Core State Standards. It is aligned to Standard G-C.2, which requires students to identify and describe relationships among inscribed angles, radii, and chords, including the tangent being perpendicular to the radius.
A tangent line touches a circle at one point and is perpendicular to the radius drawn to that point.
Theory
A tangent line touches a circle at exactly one point, the point of tangency. The central fact:
This right angle lets you form a right triangle (radius, tangent, and the line to an external point) and apply the Pythagorean theorem.
The tangent right triangle:
How to solve tangent problems
- Draw the radius to the point of tangency — it is perpendicular to the tangent.
- Form the right triangle with the radius and the tangent.
- Apply the Pythagorean theorem to find the unknown.
The radius is perpendicular to the tangent at the point of tangency.
| \(\text{angle}\) | \(=\) | \(90^\circ\) |
\(\triangle OTP\) is right-angled at \(T\); use the Pythagorean theorem.
| \(PT\) | \(=\) | \(\sqrt{OP^2-OT^2}=\sqrt{169-25}\) |
| \(=\) | \(\sqrt{144}=12\) |
Exactly one — the point of tangency.
Use the right triangle \(OTP\).
| \(OT\) | \(=\) | \(\sqrt{10^2-8^2}=\sqrt{36}=6\) |
Common pitfalls
Frequently asked questions
What is a tangent line to a circle?
A line that touches the circle at exactly one point, the point of tangency.
What is the relationship between a radius and a tangent?
The radius drawn to the point of tangency is perpendicular to the tangent line.
How do you find a tangent length from an external point?
Use the right triangle: tangent length \(=\sqrt{OP^2-r^2}\), where \(OP\) is the distance to the center.
How many points does a tangent share with a circle?
Exactly one, the point of tangency.