333. Tangent, Secant, & Point of Contact · One touch vs. two cuts

Try to break it

Drag T around the circle — the blue tangent always touches at exactly one point. Drag A around — the secant always cuts at two points (A and B). Push A all the way to T and the two cut points coincide — the secant degenerates into a tangent at the limit. That's why a tangent is the "single-touch" limit of a secant.

How you build it

Mark the centre O and draw the circle, then build a tangent that just touches at T and a secant that cuts across at A and B.

• Mark the centre O of the circle.
• With centre O, draw a circle of radius about 200.
• Mark the point of contact T on the circle.
• Draw the tangent line through T — it touches the circle at exactly one point.
• Mark a point A where the secant will cut the circle.
• Mark the second point B where the secant cuts the circle.
• Draw the secant line through A and B — it cuts the circle at two points.

The proof, step by step

Prove that a tangent meets the circle at one point and a secant at two.

• A tangent line intersects the circle at exactly one point.
• A secant line intersects the circle at exactly two points.
• The single point where a tangent touches the circle is called the point of contact.

Worked example

A line intersecting a circle in exactly two points is called a:

A secant is defined as a straight line that intersects a circle at exactly two distinct points. A tangent touches at only one point.

• Tangent
• Secant — correct
• Radius
• Chord