Fundamental Theorem

Tangent Perpendicular to Radius

Theorem 10.1
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Current Step: Welcome
Given: Circle O, Tangent XY
To Prove: OP ⊥ XY
Distance OP: 120px
Distance OQ: --
Relationship: OQ > OP
Welcome to the fundamental theorem about tangents and radii. Choose a demonstration to explore this crucial geometric relationship.
Proof by Contradiction
Step 1:
Given: Circle with center O, tangent line XY touching the circle at point P
Step 2:
To Prove: The radius OP is perpendicular to tangent XY (OP ⊥ XY)
Step 3:
Take any point Q on the tangent line XY, where Q ≠ P
Step 4:
Since XY is a tangent, point Q must lie outside the circle (if Q were inside, XY would be a secant, not a tangent)
Step 5:
Since Q is outside the circle and P is on the circle: OQ > OP (distance from center to external point > radius)
Step 6:
This means OP is the shortest distance from center O to any point on line XY
Step 7:
Conclusion: By the shortest distance principle, OP must be perpendicular to XY. Therefore: OP ⊥ XY
Real-World Applications
Bicycle Wheel Mechanics

When a bicycle wheel rolls on the ground, the ground acts as a tangent to the circular wheel. At the point of contact, the radius (spoke) to that point is always perpendicular to the ground. This principle ensures smooth rolling motion.

Engineering & Design

In mechanical engineering, this theorem is crucial for designing gears, pulleys, and rotating machinery. The perpendicular relationship ensures optimal force transmission and minimal wear.

Architecture

Architects use this principle when designing curved structures, arches, and domes. The perpendicular relationship helps determine load distribution and structural integrity.

Physics & Optics

In optics, when light rays are tangent to curved mirrors or lenses, the normal (perpendicular) to the surface determines the angle of reflection or refraction.