Tangent lines

Learning Objectives Define a tangent line and identify the point of tangency. State and apply the Tangent-Radius Theorem to solve for unknown lengths and angles. State and apply the Two Tangents from a Point Theorem to find unknown segment lengths. Use the Pythagorean theorem to verify if a line is tangent to a circle. Solve multi-step problems involving tangent lines, including finding the perimeter of polygons circumscribed about a circle. Construct a formal proof using the properties of tangent lines. Ever wondered how a bicycle chain grips the gears perfectly or how sparks fly off a grinding wheel in a straight line? ⚙️ That's the geometry of tangent lines in action! In this tutorial, we will explore one of the most important lines related to a circle: the tangent...

TermDefinitionExample CircleThe set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the center).A bicycle wheel, where the hub is the center and the spokes represent radii. Tangent LineA line in the same plane as a circle that intersects the circle at exactly one point.A straight road that just touches the edge of a circular roundabout. Point of TangencyThe single point where a tangent line touches or intersects a circle.The exact spot where a tire touches the road at any given moment. Secant LineA line that intersects a circle at two distinct points. It's important to distinguish this from a tangent line.A straight bridge passing over a circular pond, with its support pillars entering and exiting the water. Common TangentA line, ray, or segment...

Tangent-Radius Theorem A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency. This theorem is crucial because it establishes a 90° angle between the tangent line and the radius at the point of tangency. This allows you to create a right-angled triangle and use the Pythagorean theorem (a^2 + b^2 = c^2) to find missing lengths. Two Tangents from a Point Theorem If two tangent segments are drawn to a circle from the same external point, then they are congruent. Also known as the 'Hat Theorem' because the two tangent segments and the external point can look like a party hat on the circle. Use this rule to set the lengths of the two tangent segments equal to each other, which is very useful for solving algebr...