Tangent Radius Theorem
The Tangent-Radius Theorem states that the tangent at any point of a circle is perpendicular to the radius drawn through the point of contact. This fundamental geometric principle means that the angle formed between the tangent line and the radius at the exact point they meet is always 90 degrees.
Key Concepts
A tangent is a straight line that touches a circle at exactly one single point, known as the point of contact.
The shortest distance from the center of a circle to any point on the tangent line is the straight path to the point of contact.
In geometry, the shortest distance from a point to a line is always the perpendicular distance, proving that the radius and tangent meet at a right angle.
Formula & Equation
OP is perpendicular to AB (Angle OPB = 90 degrees)
O is the center of the circle, P is the point of contact, OP is the radius, and AB is the tangent line.
Common Misconceptions
Myth: A tangent line is perpendicular to every radius of the circle.
Fact: A tangent is only perpendicular to the specific radius that is drawn directly to the point of contact. It is not perpendicular to any other radius in the circle.
Real World Applications
A bicycle wheel resting on a flat road, where the road is the tangent and the vertical spoke connecting the center to the ground is the radius forming a 90-degree angle.
Sparks flying off a spinning grinding wheel travel in a straight line that is perpendicular to the radius of the wheel at the point of release.
Frequently Asked Questions
What is the Tangent-Radius Theorem?
It is a theorem stating that a tangent to a circle forms a 90-degree angle with the radius drawn to the point of contact.
Why is the radius perpendicular to the tangent?
Because the radius at the point of contact represents the shortest distance from the center of the circle to the tangent line, and the shortest distance between a point and a line is always perpendicular.