What you’ll learn
Understand radius-tangent perpendicularity.
Key idea
The perpendicular theorem concerns the radius ending at that contact point, not every radius.
Only the radius to the contact point is perpendicular to that tangent; a different radius can be parallel.
Worked example
Classify examples and non-examples for the concept 'Understand radius-tangent perpendicularity'.
Circle center O=(-3, 0) has radius 4. T=(-3, 4) is on it and line l: y=4 is tangent exactly at T. Classify the relation between radius OT and l, including where it applies.
- Contact radius vector OT=(0, 4); tangent direction (1, 0) has dot product zero with it.
- Therefore OT is perpendicular to l at T; a different radius OS=(4, 0) is parallel to l, not perpendicular.
Answer: OT perpendicular to l at T — the radius ends at the point of tangency.
Quick Check
Classify examples and non-examples for the concept 'Understand radius-tangent perpendicularity'.
Circle center O=(0, 0) has radius 2. T=(2, 0) is on it and line l: x=2 is tangent exactly at T. Classify the relation between radius OT and l, including where it applies.