Tangent Intuition Math Example 4
Follow the full solution, then compare it with the other examples linked below.
Example 4
easyAt what angle does the tangent to a circle meet the radius drawn to the point of tangency? Use this to explain why a tangent touches the circle at only one point.
Solution
- 1 Step 1: By definition, a tangent to a circle meets the radius at the point of tangency at exactly .
- 2 Step 2: If the tangent crossed the circle at a second point, that chord would have an interior right angle with the radius, but a chord inside a circle cannot be perpendicular to the radius at both endpoints simultaneously (unless it is the diameter). Therefore the line touches at exactly one point.
Answer
; the tangent meets the circle at exactly one point because the perpendicularity condition is satisfied only at one location.
The angle between radius and tangent is both the definition of tangency and the explanation for why no second intersection exists. Any other line through that point would enter the circle's interior, creating two intersection points.
About Tangent Intuition
A line that just barely touches a curve at exactly one point without crossing it, matching the curve's direction at that point.
Learn more about Tangent Intuition →More Tangent Intuition Examples
Example 1 medium
Find the equation of the tangent line to the circle [formula] at point [formula].
Example 2 hardFrom external point [formula], find the length of the tangent to the circle [formula].
Example 3 mediumVerify that the line [formula] is tangent to the circle [formula], and find the point of tangency.