Circles — Interactive

Class X · Mathematics Ch.10 · 6 Interactive Modules

Slide the line towards or away from the centre. When it is far it misses (0 points); when the distance equals the radius it is a tangent (1 point); when closer it is a secant (2 points).
The tangent touches the circle at exactly one point. Rotate the point of contact and watch the tangent line stay touching — never crossing — the circle.
Theorem 1: the radius drawn to the point of contact is always perpendicular (90°) to the tangent — no matter where the contact point is.
Theorem 2: the two tangents PA and PB from an external point P are always equal. Move P and watch both lengths stay identical.
Tangent length ℓ = √(d² − r²). The radius, tangent, and line to the centre form a right triangle — Pythagoras gives the tangent length.

🚲 Bicycle Chain

The chain runs tangent to both gear wheels — the straight sections are common tangents to two circles.

⚙️ Pulley Belts

A belt over two pulleys touches each along a tangent line, transferring rotation efficiently.

🏀 Ball on Ground

A ball resting on a flat surface touches it at one point — the ground is tangent to the ball.

📐 Inscribed Circles

A circle inscribed in a triangle or quadrilateral touches every side — each side is a tangent.

🛰️ Satellite Range

The horizon line from a satellite is tangent to the Earth, defining the edge of its coverage.

🎯 Gear Design

Meshing gears and cam followers rely on tangent contact for smooth motion.

Tangents describe how curved objects touch straight surfaces and each other — everywhere in engineering and daily life.