Lesson 2 — Tangents
Big idea: A tangent touches a circle at exactly one point, and it is perpendicular to the radius drawn to that point.
Words to know
| Word | Meaning |
|---|---|
| Tangent | A line in the plane of a circle that intersects the circle in exactly one point. |
| Point of contact | The single point where a tangent touches the circle; also called the point of tangency. |
| Radius to the point of contact | The radius drawn from the center to the point where the tangent touches. |
| Converse | The reverse statement: if a line is perpendicular to a radius at its outer endpoint, the line is tangent. |
Concepts to know
- A tangent intersects the circle in exactly one point.
- Theorem 59: If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of contact.
- Theorem 60: If a line is perpendicular to a radius at its outer endpoint, it is tangent to the circle.
- Theorems 59 and 60 are converses, so you can use either direction.
- A tangent line is 90° to the radius at the point of contact, which creates a right triangle with the center, the point of contact, and any external point on the tangent.
- The distance from the center to a tangent line equals the radius.
Formulas / rules
- No new formula. Use the Pythagorean theorem in the right triangle formed by the radius, a tangent segment, and the segment from the center to the external point:
- OP² = OA² + AP², where OA is the radius and AP is a tangent segment.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Prove a line is tangent | Show it is perpendicular to a radius at the radius’s outer endpoint. |
| Find a length involving a tangent | Mark the right angle at the point of contact, then use the Pythagorean theorem. |
| Find a radius | Use the right triangle and subtract the tangent segment’s square from the external segment’s square. |
| Identify the point of contact | It is the one point shared by the line and the circle. |
| Explain why a line is not tangent | Show it meets the circle in two points, or is not perpendicular to the radius at the endpoint. |
| Solve a real-world tangent problem | Sketch the circle, the tangent, and the radius to the contact point. |
Watch out
- A line that merely looks like it touches the circle is not necessarily tangent; it must meet the circle in exactly one point.
- The right angle is at the point of contact, not at the circle’s center.
- The radius and tangent meet at the endpoint of the radius, not somewhere inside it.
- Don’t assume every line perpendicular to a chord is tangent; it must be perpendicular to a radius at its outer endpoint.
- Keep the tangent segment and the radius as the legs; the segment from the center to the external point is the hypotenuse.