A tangent line is a line that touches a circle at exactly one point, called the point of tangency. Tangents are important because they connect straight-line geometry with circle geometry in a precise way. They appear in construction problems, proofs, coordinate geometry, and real-world designs involving wheels, pulleys, arcs, and sight lines.
The key idea is that a tangent only just touches the circle instead of cutting through it.
The radius drawn to the point of tangency is always perpendicular to the tangent line. This creates a right triangle when a tangent is drawn from an external point to a circle, which lets you use the Pythagorean theorem. If two tangent segments are drawn from the same external point, those segments have equal lengths.
These properties make tangent diagrams powerful tools for finding missing lengths, proving relationships, and recognizing right angles.
Key Facts
- A tangent line touches a circle at exactly one point.
- If line PT is tangent to a circle at T, then OT is perpendicular to PT.
- The radius to the point of tangency forms a right angle with the tangent: angle OTP = 90 degrees.
- Tangent segments from the same external point are congruent: PA = PB.
- For external point P and tangent point T, OP^2 = OT^2 + PT^2.
- A secant intersects a circle at two points, while a tangent intersects it at one point.
Vocabulary
- Tangent line
- A line that intersects a circle at exactly one point.
- Point of tangency
- The single point where a tangent line touches a circle.
- Radius
- A segment from the center of a circle to any point on the circle.
- External point
- A point outside a circle from which tangents or secants can be drawn.
- Congruent segments
- Segments that have exactly the same length.
Common Mistakes to Avoid
- Assuming the tangent is perpendicular to any line through the tangency point, which is wrong because it is only guaranteed perpendicular to the radius drawn to that point.
- Thinking a tangent crosses the circle at two points, which is wrong because that describes a secant line, not a tangent line.
- Using PA = PB for segments drawn from different external points, which is wrong because tangent segments are equal only when they share the same external point.
- Forgetting to make a right triangle with the radius and tangent segment, which is wrong because the 90 degree angle is the reason the Pythagorean theorem applies.
Practice Questions
- 1 A circle has center O and radius OT = 6 cm. Point P is outside the circle, and PT is tangent at T. If OP = 10 cm, find PT.
- 2 From external point P, two tangents PA and PB are drawn to a circle. If PA = 3x + 2 and PB = 5x - 10, find x and the length of each tangent segment.
- 3 A student says that if a line touches a circle at point T, then it must be perpendicular to every chord passing through T. Explain why this is not correct and identify the segment that is guaranteed to be perpendicular to the tangent.