Tangent and secant

When a line and a circle are drawn together in the plane, exactly three things can happen. The line might miss the circle entirely (no common points). It might cut through the circle, meeting it at two distinct points , this is a secant. Or it might just kiss the circle at exactly one point , this is a tangent, and the meeting point is called the point of contact.

The intuition is best built by an experiment. Draw a circle and a line close to it but not touching. Now slide the line slowly toward the circle. At first there is a gap; then the line touches the circle at exactly one point (tangent); then it cuts the circle into a chord with two endpoints (secant); then it cuts a longer chord; then a chord that is a diameter; and so on. The tangent is the boundary case , the moment between "missing" and "cutting".

Definitions

A secant to a circle is a line that intersects the circle in two distinct points. A tangent to a circle is a line that meets the circle in exactly one point. That single point is the point of contact (or point of tangency).

A chord is the line segment between the two intersection points of a secant; a diameter is a chord that passes through the centre.

How many tangents exist from a point?

This is a useful classification, depending on where the point lies relative to the circle:

• Point inside the circle: no tangent can be drawn from it (every line through it cuts the circle in two points).
• Point on the circle: exactly one tangent (the unique tangent at that point).
• Point outside the circle: exactly two tangents.

The "exactly two" claim for an external point is one of the most-used facts of this chapter. It immediately raises the question , do those two tangents have the same length? They do (Theorem 2, the next subtopic).

Tangent as a limit of secants

Think of a secant cutting the circle at two points $A$ and $B$. Now let $B$ slide along the circle toward $A$. As $B$ approaches $A$, the line $AB$ keeps cutting the circle in two points, but those points are getting closer together. In the limit, when $B$ coincides with $A$, the secant has become the tangent at $A$.

This view explains why the tangent at $A$ is unique , it is the unique limiting position of secants through $A$ as the second intersection point approaches $A$. It also gives a geometric reason for the tangent-radius perpendicularity theorem (next subtopic): in the limit, the chord $AB$ shrinks to a point, and the limiting "direction" of the chord becomes the tangent direction.

Tangent-radius perpendicularity

Theorem. The tangent at any point of a circle is perpendicular to the radius drawn to that point.

Proof sketch (by contradiction). Let $\ell$ be the tangent at $P$ to a circle of centre $O$. Suppose $OP$ is not perpendicular to $\ell$. Then drop a perpendicular $OQ$ from $O$ to $\ell$, with $Q \neq P$ a point on $\ell$. Now $\triangle OQP$ is right-angled at $Q$, so $OP > OQ$ (the hypotenuse is the longest side). This says the distance from $O$ to $Q$ is less than the radius, so $Q$ is inside the circle. But $Q$ lies on $\ell$, contradicting that $\ell$ meets the circle in only one point. Hence the assumption fails, and $OP \perp \ell$. $\blacksquare$

The contrapositive is also useful: if you have a line through a point $P$ on a circle, and that line happens to be perpendicular to the radius $OP$, then it is the tangent at $P$.

Worked examples

Example 1. A circle of centre $O$ has radius $5$ cm. A line passes at distance $5$ cm from $O$. State whether it is a tangent, secant, or neither.

If the perpendicular distance from $O$ to the line equals the radius, then the line is tangent to the circle. Answer: tangent.

Example 2. A point $P$ is at distance $13$ cm from the centre $O$ of a circle of radius $5$ cm. Find the length of a tangent from $P$ to the circle.

The tangent from $P$ touches the circle at some point $A$. Then $OA = 5$ (radius), $OP = 13$. Since $OA \perp PA$, by Pythagoras, $PA^2 = OP^2 - OA^2 = 169 - 25 = 144$, so $PA = 12$ cm.

Example 3. Prove that the tangent to a circle at any point is perpendicular to the radius at that point.

(Use the proof sketch above. In a board exam, present it as a proper proof: state the setup, the assumption for contradiction, the deduction, and the contradiction.)

Example 4. A circle of radius $6$ cm has a tangent at point $A$. From $A$, drop a perpendicular to the tangent , what is its length?

The tangent is along one direction; the perpendicular to the tangent at $A$ is in the direction $OA$. The radius is $6$ cm.

Example 5. A line is at distance $3$ cm from the centre of a circle of radius $4$ cm. How many points of intersection?

Since $3 < 4$, the line cuts the circle in two points (it is a secant). The chord cut off has length $2\sqrt{4^2 - 3^2} = 2\sqrt 7$ cm.

Try it yourself

1. State the definitions of tangent, secant, chord, and diameter.
2. How many tangents pass through a point inside a circle?
3. A line is at distance $7$ cm from the centre of a circle of radius $5$ cm. Tangent, secant, or neither?
4. A point is $10$ cm from the centre of a circle of radius $6$ cm. Find the length of the tangent from the point.
5. Prove: if a line through $P$ on a circle is perpendicular to the radius $OP$, it is tangent at $P$.
6. From an external point at distance $25$ from a centre of a circle of radius $7$, find the tangent length.
7. Two tangents are drawn from a point $T$ to a circle of radius $5$ cm; the tangent length is $12$ cm. Find $OT$.
8. A circle has a tangent at $P$ and a chord $PQ$. The chord makes angle $30^\circ$ with the tangent. Find the angle $\angle OPQ$.
9. Show that the perpendicular from the centre to a chord bisects the chord.
10. From a point on a circle, the perpendicular distance to the tangent at any other point is at most the diameter. Why?
11. State and prove the theorem of tangent-radius perpendicularity.
12. A wheel of radius $30$ cm rolls without slipping on the ground. At the moment of contact, what is the angle between the ground and the radius to the contact point?

Pitfalls / Insight

A frequent mistake is to call a chord a tangent. A tangent must meet the circle in exactly one point; a chord meets it in two. Another mistake: assuming that any line through a point on the circle is tangent , only the unique perpendicular to the radius at that point qualifies.

The deepest insight: when you see a tangent in a problem, immediately draw the radius to the point of contact and mark the right angle. Then look for Pythagoras. That single instinct unlocks most board questions on this chapter.