Class 10 · Chapter 10 · Interactive Lab
A rope over a pulley, a bicycle wheel touching the road — both are tangents to a circle. Drag things around in each panel below to see exactly when a line touches a circle, why the touch is always at a right angle, and what stays equal no matter where you stand.
Slide the vertical line PQ toward the circle. Compare its distance d from the centre with the radius r: three possibilities, exactly as in Fig. 10.1.
drag the slider to move line PQ toward the circle
Drag point P around the circle to choose the point of contact — the tangent line there is always perpendicular to radius OP. Then drag point Q along that tangent line: OQ is always longer than OP, except exactly at P. That's the proof idea: OP is the shortest distance from O to the tangent line, which is only possible if OP ⊥ tangent.
● P sets the point of contact · ● Q slides along the tangent
Drag the coral point P anywhere on the canvas. Inside the circle: no tangent is possible. On the circle: exactly one. Outside: exactly two — and Theorem 10.2 says those two tangent lengths, PT₁ and PT₂, are always equal.
drag point P anywhere
Slide the four points of contact P, Q, R, S around the circle. Each vertex of quadrilateral ABCD sits where two tangent lines meet, so the two tangent segments from that vertex are always equal — and that forces AB + CD = AD + BC, no matter how you drag the points.
four points of contact, one on each side