Tangents

Intuition and core definition

A tangent line to a circle meets it at exactly one point of tangency and is perpendicular to the radius drawn to that point. Tangent segments from the same exterior point are congruent. These properties connect right triangles, power of a point, and circle proofs.

Notation, language, and conditions

If $PT$ is tangent at $T$ to a circle centred $O$, then $OT\perp PT$. From exterior point $P$ with tangency points $A,B$, $PA=PB$. A tangent–chord angle equals half its intercepted arc under the tangent-chord theorem.

Why this idea matters

A tangent meets a circle at one point and is perpendicular to the radius drawn to that point of contact.

A dependable method

1. Locate the centre, exterior point, and claimed contact point.
2. Draw the radius to the contact point and mark the right angle.
3. Use the Pythagorean theorem for tangent length or equal-tangent theorem for paired segments.
4. Apply angle/arc relations only after identifying the intercepted arc.
5. Check positivity and that the tangent segment lies outside except at contact.

Worked example

Common mistakes