Tangent lines - Geometry

1. Fundamental Concepts

Definition: A tangent line to a circle is a line that touches the circle at exactly one point, known as the point of tangency.
Key Property: The radius of the circle is perpendicular to the tangent line at the point of tangency.
Tangent-Secant Theorem: If a tangent and a secant are drawn from an external point to a circle, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external part.

Tangent Line Equation: $$(x - x_1)^2 + (y - y_1)^2 = r^2$$

Tangent-Secant Theorem: $$t^2 = s(s + e)$$

Application: Used in various geometric proofs and constructions

Problem: Find the equation of the tangent line to the circle $$x^2 + y^2 = 9$$ at the point $$(3, 0)$$.

The center of the circle is $$(0, 0)$$ and the radius is $$3$$. The slope of the radius at $$(3, 0)$$ is undefined (vertical).
The tangent line will be horizontal with the equation $$y = 0$$.

Validation: Substitute $$(3, 0)$$ into the circle’s equation → Original: $$3^2 + 0^2 = 9$$; Simplified: $$9 = 9$$ ✓

Problem: Given a circle with center $$(2, 3)$$ and radius $$5$$, find the equation of the tangent line at the point $$(7, 8)$$.

Find the slope of the radius: $$\frac{8 - 3}{7 - 2} = \frac{5}{5} = 1$$.
The slope of the tangent line is the negative reciprocal of the radius slope, which is $$-1$$.
Using the point-slope form of the line equation: $$y - 8 = -1(x - 7)$$.
Simplify to get the equation of the tangent line: $$y = -x + 15$$.

Validation: Substitute $$(7, 8)$$ into the tangent line equation → Original: $$8 = -7 + 15$$; Simplified: $$8 = 8$$ ✓

Visual Strategy: Draw the circle and the tangent line to visualize the relationship between the radius and the tangent.
Error-Proofing: Always check if the point of tangency satisfies both the circle’s equation and the tangent line’s equation.
Concept Reinforcement: Practice identifying the slope of the radius and using it to find the slope of the tangent line.