Secant of a Circle — Definition, Formula, and Examples
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Secant of a Circle — Definition, Formula, and Examples
TL;DR
A secant of a circle is a straight line that intersects the circle at two distinct points — it is a chord extended past both ends. This article covers the secant definition, how it differs from a chord and a tangent, the two power-of-a-point theorems (PA⋅PB=PC⋅PD) and the tangent–secant relation, and worked examples of a secant of a circle.
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 PPP, draw two secants. The first crosses the circle at AAA (near) and BBB (far); the second at CCC (near) and DDD (far). Then:
PA⋅PB=PC⋅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 PPP, draw one tangent touching at TTT and one secant crossing at AAA (near) and BBB (far). Then:
PT²=PA⋅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 |
|---|---|
| PPP | The external point where the lines meet |
| PA, PC | Distance from PPP to the near intersection |
| PB, PD | Distance from PPP to the far intersection |
| PT | Length of the tangent segment from PPP to the point of contact |
Examples of Secant of a Circle
Example 1
A line meets a circle at points MMM and NNN 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 PPP, 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⋅PD=3×6=18, while the first is PA⋅PB=4×9=36, so the two products are not equal and the subtraction approach is wrong.
The correct method uses the two-secant theorem:
PA⋅PB=PC⋅PD → 4×9=3×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⋅PB=PC⋅PD → 5×12=6×PD → PD=10.
Final answer: PD=10.
Example 4
From a point PPP outside a circle, a tangent of length PT=8 touches the circle, and a secant from PPP has near point A with PA=4. Find the whole secant length PB.
Use the tangent–secant theorem:
PT²=PA⋅PB → 8²=4×PB → PB=16.
Final answer: PB=16.
Example 5
A secant from external point PPP 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:
PT²=PA⋅PB → PT²=3×8 → PT=√24 ≈ 4.9.
Final answer: PT≈4.9.
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:
∠P=0.5(far arc−near arc) → ∠P=0.5(110°−40°) → ∠P=35°.
Final answer: ∠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.
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⋅AB=PC⋅CD using the interior chord pieces.
The correct way: Each side of PA⋅PB=PC⋅PD is (external segment) × (whole secant).
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.
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 PT²=AB² or PT=PA⋅PB without the square.
The correct way: It is PT²=PA⋅PB — the tangent length is squared.
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⋅PB=PC⋅PD from an external point.
The tangent–secant theorem gives PT²=PA⋅PB.
These are product relationships, not differences — and they describe the circle from a point outside it.
Frequently Asked Questions
Can a secant of a circle pass through the center?
Yes. A secant of a circle through the center contains the diameter — the diameter is the chord on that secant, and it is the longest possible chord.
Is every chord part of a secant?
Yes. Extend any chord beyond both endpoints and you get the secant that contains it.
How many times can a straight line cross a circle?
At most twice. A line can miss the circle entirely (no intersection), touch it once (a tangent), or cross it twice (a secant). It can never meet a circle at three points.
What is the secant line in calculus — is it the same thing?
The names are related. In calculus, a secant line joins two points on a curve, and as those points slide together it becomes the tangent line — the same merging idea you see when a circle's secant collapses into a tangent.