Tangent Lines

Geometry · Axiom Academy

Lines that touch a circle at exactly one point — and the right angle hiding at that point. Picture a straight line drifting toward a circle. Far away it misses entirely; pushed in too far it becomes a secant , crossing the circle at two points. Exactly in between is one special position where the line meets the circle at a single point — that line is the tangent , and the point it touches is the point of tangency . 2. The Key Property: Radius ⊥ Tangent Here is the fact that makes tangent problems solvable. Draw the radius to the point of tangency, then watch it as the point travels around the circle. No matter where the contact point sits, the radius and the tangent line always meet at a perfect 90° . The radius drawn to the point of tangency is perpendicular to the tangent. 3. Properties of Tangent Lines Everything else about tangents follows from that single right angle. Four properties worth knowing: The tangent meets the radius at the point of tangency at exactly 90°. Two tangent segments drawn from the same external point are congruent. A tangent touches the circle at exactly one point — it never crosses inside. The tangent, the radius, and the segment from the center to an external point form a right triangle. A line can be tangent to two circles at once — a common tangent . There are two kinds, told apart by whether the line passes between the circles. Do not cross between the two circles — they stay on the outside.

This is the written version of the interactive lesson above. See the full Geometry course.