Tangents

Tangent properties support circle construction, optics, belt design, and exact distance problems. They turn contact into perpendicular structure.

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

Representations and interpretation

The radius-to-tangent right triangle gives a geometric test for tangency. From one exterior point, congruent right triangles share hypotenuse $OP$ and have equal radii, explaining equal tangent lengths.

Reasoning about variations

A secant crosses the circle at two points and is not perpendicular to a radius in general. A line that merely looks as though it touches requires exact distance or perpendicular evidence.

Common mistakes

How to check your work

• Verify the centre-to-line distance equals radius.
• Check the right-triangle equation.
• Compare two tangent lengths from the same exterior point.

Practice

1. A point is $10$ from a circle centre and radius is $6$. Find tangent length.
2. What angle does a tangent make with the radius at contact?
3. Two tangents from point $P$ have lengths $3x+1$ and $5x-7$. Find $x$.

Answers and brief solutions

Synthesis and transfer

A straight path just touching a circular exclusion zone forms a right triangle with the centre, allowing the tangent length to be found and checked against the hypotenuse.

Let an external point be $13$ units from a circle's centre and let the radius be $5$. The radius to the contact point is perpendicular to the tangent segment, so its length is $\sqrt{13^2-5^2}=12$. Two tangent segments drawn from the same external point have equal length, a fact that can be proved by congruent right triangles sharing the centre-to-external hypotenuse. If the external point moves onto the circle, tangent length shrinks to zero; inside the circle, no real tangent segment exists. These boundary cases agree with the square-root expression $\sqrt{d^2-r^2}$.

Related topics

• Circle Terminology
• Chords
• Pythagorean Theorem

Teaching and accessibility note