lesson
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Spin a ball on a string and suddenly let go: it doesn't fly straight out from the center, but shoots off along a straight path that grazes the circle at that exact instant.
The Point of Tangency
A line in the same plane that touches a circle at exactly one location is a tangent line, and that single contact spot is the point of tangency.
📊A clean interactive-style geometry diagram on light gray background (#f8f9fa). Shows a circle with center O, a radius segment drawn in blue (#2563eb) to point P on the circle, and a bold orange (#ea580c) tangent line t grazing the circle precisely at P. Labels: 'Center O', 'Radius r', 'Point of Tangency P', 'Tangent Line t'. A smooth pulsing indicator at point P highlights that there is only one shared point between the line and the circle. Responsive card format with rounded corners.
Around 300 BCE, the Greek mathematician Euclid proved in Book III of his Elements that this grazing line always forms an exact 90∘ angle with the radius drawn to that contact spot.
Tangent-Radius Perpendicularity
If a line is tangent to a circle, then it is perpendicular (meets at 90∘) to the radius drawn to the point of tangency.
📊Diagram showing a circle with center O and a tangent line touching at point T. A radius OT is drawn in navy (#1e293b). At point T, a prominent square right-angle symbol is highlighted in green (#16a34a) with the label '90°'. A math callout box reads: 'Line AB ⊥ Radius OT at Point T'. Clean labels, high contrast, responsive container (max-width 350px).
Why does this contact point always produce a square corner instead of slanting at some other angle?
Why It Works
The shortest distance from any point to a straight line is always the perpendicular segment directly connecting them.
Because the tangent line only touches the circle at point P, every other point Q on the line lies outside the circle, meaning distance OQ>OP. That makes radius OP the unique shortest path to the line, so it must be perpendicular.
📊Visual geometric proof diagram. A circle with center O. Radius OP goes to tangent line at P (length r). Another point Q on the tangent line is outside the circle, connected to center O by a dashed orange line labeled 'Distance > r'. A text annotation explains: 'OP is the shortest distance from O to the line → OP must be perpendicular to the tangent.' Clean minimalist layout.