What Is A Secant Of A Circle?
A secant of a circle is a straight line that intersects the circle at exactly two distinct points. Because it is a full line, not a segment, it extends infinitely in both directions past the circle.
The key contrast is with a chord, which is the segment whose two endpoints lie on the circle. Take a chord and extend it both ways and you have a secant — the secant contains the chord. A tangent is the limiting case: slide the two intersection points of a secant together until they merge into one, and the secant becomes a tangent touching the circle at a single point.
How Is A Secant Different From A Chord And A Tangent?
This is the question students ask most, because the three terms describe lines that look almost the same on a quick sketch. The difference is purely about how many times the line meets the circle and whether it stops.
Line | Meets the circle at | Extends beyond the circle? |
|---|---|---|
Chord | 2 points | No — it stops at the circle |
Secant | 2 points | Yes — it continues past both points |
Tangent | 1 point | Yes — it touches and continues |
A chord and a secant share the same two intersection points; the chord is just the bounded piece in the middle. A tangent meets the circle only once and sits perpendicular to the radius at that point.
The secant theorems and their formulas
Secants matter because of what happens at the point where two of them meet outside the circle. These relationships are forms of the power of a point — a result first studied through the work of Jakob Steiner in the 1820s.
Two-secant (intersecting secants) theorem. From an external point $P$, draw two secants. The first crosses the circle at $A$ (near) and $B$ (far); the second at $C$ (near) and $D$ (far). Then:
$$PA \cdot PB = PC \cdot PD$$
Each product multiplies the whole secant length by its external part, and the two products are equal.
Tangent–secant theorem. From an external point $P$, draw one tangent touching at $T$ and one secant crossing at $A$ (near) and $B$ (far). Then:
$$PT^2 = PA \cdot PB$$
The tangent length squared equals the secant's external part times its whole length. This is the two-secant theorem with the tangent treated as a secant whose two points have merged.
Symbol | Meaning |
|---|---|
$P$ | The external point where the lines meet |
$PA$, $PC$ | Distance from $P$ to the near intersection |
$PB$, $PD$ | Distance from $P$ to the far intersection |
$PT$ | Length of the tangent segment from $P$ to the point of contact |
Examples of Secant of a Circle
The examples build from naming a line to applying both theorems. Each step is on its own line.
Example 1
A line meets a circle at points $M$ and $N$ and continues past both. Is it a chord, a secant, or a tangent?
The line meets the circle at two points and extends beyond them. A chord would stop at the circle; a tangent would touch only once.
Final answer: it is a secant.
Example 2
From an external point $P$, two secants give $PA = 4$, $PB = 9$, and $PC = 3$; a student finds $PD = 9 - 3 = 6$, so find the correct $PD$.
The first instinct is to subtract, treating the segments as if they simply add and remove along one line. Test it: with $PD = 6$, the second product is $PC \cdot PD = 3 \times 6 = 18$, while the first is $PA \cdot PB = 4 \times 9 = 36$, so the two products are not equal and the subtraction approach is wrong.
The correct method uses the two-secant theorem:
$$PA \cdot PB = PC \cdot PD$$ $$4 \times 9 = 3 \times PD$$ $$36 = 3 \times PD$$ $$PD = 12$$
Final answer: $PD = 12$.
Example 3
Two secants from an external point give $PA = 5$, $PB = 12$, and $PC = 6$. Find $PD$.
Apply the two-secant theorem:
$$PA \cdot PB = PC \cdot PD$$ $$5 \times 12 = 6 \times PD$$ $$60 = 6 \times PD$$ $$PD = 10$$
Final answer: $PD = 10$.
Example 4
From a point $P$ outside a circle, a tangent of length $PT = 8$ touches the circle, and a secant from $P$ has near point $A$ with $PA = 4$. Find the whole secant length $PB$.
Use the tangent–secant theorem:
$$PT^2 = PA \cdot PB$$ $$8^2 = 4 \times PB$$ $$64 = 4 \times PB$$ $$PB = 16$$
Final answer: $PB = 16$.
Example 5
A secant from external point $P$ crosses a circle at $A$ and $B$ with $PA = 3$ and $AB = 5$. Find the length of the tangent $PT$ from the same point.
First find the whole secant length $PB$:
$$PB = PA + AB = 3 + 5 = 8$$
Then apply the tangent–secant theorem:
$$PT^2 = PA \cdot PB$$ $$PT^2 = 3 \times 8$$ $$PT^2 = 24$$ $$PT = \sqrt{24} = 2\sqrt{6} \approx 4.9$$
Final answer: $PT = 2\sqrt{6} \approx 4.9$ units.
Example 6
An angle formed by two secants meeting outside a circle equals half the difference of the two intercepted arcs. If the far arc is $110°$ and the near arc is $40°$, find the angle at the external point.
The external-angle rule for two secants is:
$$\angle P = \frac{1}{2}(\text{far arc} - \text{near arc})$$ $$\angle P = \frac{1}{2}(110° - 40°)$$ $$\angle P = \frac{1}{2}(70°)$$ $$\angle P = 35°$$
Final answer: $\angle P = 35°$.
Why Secants Are Worth Defining Separately
A natural question: if a secant is just an extended chord, why give it its own name? Because the external point is where the useful mathematics lives. A chord tells you about the inside of the circle; a secant lets you reason about a point sitting outside it, using only lengths you can measure from that point.
That is exactly the situation in real measurement. A surveyor standing outside a circular structure, or an engineer aiming a sightline that grazes a curved tank, works from an external point and reads off distances along straight sightlines. The power-of-a-point relationship turns those straight-line distances into facts about the circle they never directly touch. The same reasoning underlies optical setups where a sightline cuts a circular lens or aperture at two points.
Slip-Free Secant Work: The Mistakes To Watch
Mistake 1: Multiplying the wrong segment lengths
Where it slips in: Applying the two-secant theorem when the problem gives the outside piece and the chord piece separately, not the full length.
Don't do this: Write $PA \cdot AB = PC \cdot CD$ using the interior chord pieces.
The correct way: Each side of $PA \cdot PB = PC \cdot PD$ is (external segment) × (whole secant). $PB$ is the full distance from $P$ to the far point — add the near distance and the chord if needed.
Mistake 2: Subtracting instead of using the product relationship
Where it slips in: When the unknown looks like it could be found by simple addition or subtraction along the line.
Don't do this: Treat $PA$, $PB$, $PC$, $PD$ as if they combine linearly.
The correct way: The relationship is multiplicative — equal products, not equal sums. The rusher who reaches for subtraction gets an answer that fails the product check. The first-instinct error here is exactly that: reading a product relationship as if it were a difference, because the segments sit along straight lines that look additive.
Mistake 3: Confusing the tangent–secant form
Where it slips in: Mixing the tangent length with the full secant on the wrong side.
Don't do this: Write $PT^2 = AB^2$ or $PT = PA \cdot PB$ without the square.
The correct way: It is $PT^2 = PA \cdot PB$ — the tangent length is squared, and it equals the external part times the whole secant. The memorizer who recalls "tangent equals secant product" but forgets the square gets the units wrong; squaring is what makes both sides an area.
Conclusion
A secant of a circle is a line that cuts the circle at two distinct points and extends beyond both — a chord extended.
A chord stops at the circle, a secant continues past it, and a tangent meets the circle once.
The two-secant theorem gives $PA \cdot PB = PC \cdot PD$ from an external point.
The tangent–secant theorem gives $PT^2 = PA \cdot PB$.
These are product relationships, not differences — and they describe the circle from a point outside it.
Practice and a next step
Practice these problems to solidify your understanding: redraw each setup, label the near and far points clearly, and apply the matching theorem. If the two-secant products do not come out equal, recheck which length is the external part and which is the whole secant.
Want a live Bhanzu trainer to walk through more secant problems? Book a free demo class.
Was this article helpful?
Your feedback helps us write better content