Tangent to a Circle
GeometryA tangent to a circle is a line that touches the circle at exactly one point (the point of tangency) and is perpendicular to the radius at that point.
Formula
\text{tangent length from external point } P: t = \sqrt{d^2 - r^2}
Definition
A tangent line touches a circle at exactly one point, called the point of tangency. It does not cross through the circle - it just barely touches it at one spot.
Example
A ball resting on a flat floor: the floor is tangent to the ball at the one point where they touch. The word "tangent" comes from Latin "tangere" meaning "to touch."
Key Insight
A tangent is always perpendicular (at a right angle) to the radius drawn to the point of tangency. This perpendicularity is the key property used to solve tangent problems and construct tangent lines.
Definition
A tangent to a circle is a line (or line segment) that intersects the circle at exactly one point. The tangent is perpendicular to the radius at the point of tangency. From an external point $P$ at distance $d$ from center, the tangent length is $t = \sqrt{d^2 - r^2}$. Two tangents from an external point are equal in length.
Example
External point $P$ is $13$ units from center $O$, radius $= 5$. Tangent length $= \sqrt{169 - 25} = \sqrt{144} = 12$. The angle between the two tangents from $P$ and the line $PO$ creates a right triangle with legs $12$ and $5$, hypotenuse $13$. Check: $5^2 + 12^2 = 169 = 13^2$.
Key Insight
The equal tangent lengths from an external point (two tangents drawn to a circle are equal) is used in the proof that a circle can be inscribed in a triangle: the tangent lengths from each vertex are equal on both sides, making the incircle tangent to all three sides.
Definition
A tangent to a circle at point $T$ is the limit of secant lines through $T$ as the second intersection point approaches $T$. Equivalently, the tangent at $T$ is perpendicular to the radius $OT$. For the unit circle, the tangent at $(\cos\theta, \sin\theta)$ has equation $x\cos\theta + y\sin\theta = 1$. In differential geometry, the tangent line is the first-order approximation of the curve at $T$.
Example
Tangent to circle $x^2+y^2=r^2$ at point $(x_0,y_0)$: $xx_0 + yy_0 = r^2$. For circle $x^2+y^2=25$ at point $(3,4)$: $3x+4y=25$. Check: the point $(3,4)$ satisfies $3\cdot3+4\cdot4=25$. The perpendicularity: radius direction $(3,4)$, tangent direction $(-4,3)$: dot product $= -12+12=0$.
Key Insight
The tangent line as the limit of secant lines is the geometric definition of the derivative: the slope of the tangent equals $dy/dx$ at that point. This connection between tangent lines to curves and derivatives is the foundation of differential calculus - Leibniz and Newton both used geometric tangent constructions to motivate their calculus definitions.