Tangents to Circles

In geometry, a line that intersects a circle exactly once; in calculus, a line that touches a curve at one point and whose slope is equal to that of the curve at that point. Particularly useful as approximations of curves in the immediate vicinity of the point of tangency, tangent lines are the basis of many estimation techniques, including linear approximation. The numerical value of the slope of the tangent line to the graph of a function at any point equals that of the function's derivative at that point.A line sharing a common point with a curve or surface and being the closest linear approximation of the curve or surface at that point. In geometry, a line that intersects a circle exactly once; in calculus, a line that touches a curve at one point and whose slope is equal to that of the curve at that point. Particularly useful as approximations of curves in the immediate vicinity of the point of tangency, tangent lines are the basis of many estimation techniques, including linear approximation. The numerical value of the slope of the tangent line to the graph of a function at any point equals that of the function's derivative at that point.

A tangent line t to a circle C intersects the circle at a single point T. For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections. The tangent line through a point P on the circle is perpendicular to the diameter passing through P. Tangent to a circle is a line which intersects the circle in exactly one point.

At a point of a circle there is one and only one tangent.

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

The lengths of tangents drawn from an external point to a circle are equal.

• Centre of the circle lies on the bisector of the angle between the two tangents. Tangent to a circle is a line which intersects the circle in exactly one point. A tangent touches a circle in exactly one place
• The tangent intersects the circle's radius at a 90° angle .

Since a tangent only touches the circle at exactly one and only one point, that point must be perpendicular to a radius. To test out the interconnected relationship of these two defining traits of a tangent, try the interactive exercise below. It's only when the line is tangent to the circle that the radius will hit that line at exactly one point and at this point the line segment or tangent must intersect with the radius at a 90° angle.

The point where the tangent and the circle intersect is called the tangent of a circle. A tangent to a circle is a straight line, in the plane of the circle, which touches the circle at only one point. The point is called the point of tangency or the point of contact.

The radius, which passes through the point of contact, is perpendicular to the tangent.