Tangent-Secant Theorem: PA × PB = PT² | Circles

Quick answer: If from an external point $P$ a secant cuts a circle at $A$ and $B$, and a tangent touches it at $T$, then $PA \times PB = PT^2$. This is the tangent–secant theorem (also called the power of a point). For example, if $PA = 4\text{ cm}$ and $AB = 5\text{ cm}$, then $PB = 9\text{ cm}$ and $PT = \sqrt{4 \times 9} = 6\text{ cm}$. Inside a circle, for two chords $AB$ and $CD$ meeting at $P$: $PA \times PB = PC \times PD$.

From a point outside a circle, a line can either touch the circle at one point (a tangent) or cut it at two points (a secant). The tangent–secant theorem links their lengths with a simple equation. A similar result holds for two chords that cross inside a circle.

Syllabus note: The CBSE Class 10 NCERT chapter Circles covers tangents to a circle (for example, that the tangent is perpendicular to the radius, and that the two tangents from an external point are equal). The tangent–secant theorem, the alternate segment theorem and the intersecting chords theorem are not part of the Class 10 syllabus. This article is enrichment material.

What You Will Learn

• What tangents and secants are
• The statement of the tangent–secant theorem, $PA \times PB = PT^2$
• A proof using similar triangles
• The intersecting chords theorem, $PA \times PB = PC \times PD$
• Two solved examples

1. The Theorem

Tangent–secant theorem: If a secant from an external point $P$ meets a circle at $A$ and $B$ (with $A$ nearer to $P$), and a tangent from $P$ touches the circle at $T$, then $PA \times PB = PT^2$

          T
         /
        /  (circle through T, A, B)
       /
  P --A--------B
  PA = outer part of the secant
  PB = whole secant (P to B)
  PT = tangent

2. Proof Using Similar Triangles

Construction: Join $TA$ and $TB$.

Step 1: Use the alternate segment theorem. The angle between a tangent and a chord at the point of contact equals the angle in the alternate segment. So:

$\angle PTA = \angle PBT$

Step 2: Compare triangles $PAT$ and $PTB$.

• $\angle TPA = \angle BPT$ (common angle at $P$)
• $\angle PTA = \angle PBT$ (from Step 1)

So, by the AA similarity criterion, $\triangle PAT \sim \triangle PTB$.

Step 3: Write the ratio of corresponding sides.

$\frac{PA}{PT} = \frac{PT}{PB}$

Step 4: Cross-multiply.

$PA \times PB = PT^2$

Hence proved.

3. The Intersecting Chords Theorem

Intersecting chords theorem: If two chords $AB$ and $CD$ of a circle meet at a point $P$ inside the circle, then $PA \times PB = PC \times PD$

Proof outline: Join $AC$ and $BD$. In triangles $PAC$ and $PDB$:

• $\angle APC = \angle DPB$ (vertically opposite angles)
• $\angle PAC = \angle PDB$ (angles in the same segment, both standing on arc $BC$)

So $\triangle PAC \sim \triangle PDB$, which gives $\frac{PA}{PD} = \frac{PC}{PB}$, i.e. $PA \times PB = PC \times PD$.

4. Example 1: Finding the Tangent

Question

From an external point $P$, a secant $PAB$ meets a circle at $A$ and $B$. If $PA = 4\text{ cm}$ and $AB = 5\text{ cm}$, find the length of the tangent $PT$.

Solution

Step 1: Find the whole secant.

$PB = PA + AB = 4 + 5 = 9\text{ cm}$

Step 2: Apply the theorem.

$PT^2 = PA \times PB = 4 \times 9 = 36 \implies PT = 6\text{ cm}$

Answer: $PT = 6\text{ cm}$.

5. Example 2: Intersecting Chords

Question

Chords $AB$ and $CD$ of a circle meet at an interior point $P$. If $PA = 3\text{ cm}$, $PB = 8\text{ cm}$ and $PC = 4\text{ cm}$, find $PD$ and the length of $CD$.

Solution

Step 1: Apply the theorem.

$PA \times PB = PC \times PD \implies 3 \times 8 = 4 \times PD \implies PD = 6\text{ cm}$

Step 2: Find $CD$.

$CD = PC + PD = 4 + 6 = 10\text{ cm}$

Answer: $PD = 6\text{ cm}$ and $CD = 10\text{ cm}$.

6. Summary Table

7. Tips and Common Mistakes

Tip: $PB$ is the whole secant, from $P$ to the far point $B$: $PB = PA + AB$.

Common Mistake: Writing $PA \times AB = PT^2$. The correct statement is $PA \times PB = PT^2$.

Common Mistake: Using the theorem for a point inside the circle. Inside the circle, use the intersecting chords form, $PA \times PB = PC \times PD$.

Frequently Asked Questions

What is the tangent–secant theorem?

If a secant from an external point P meets a circle at A and B, and a tangent from P touches it at T, then PA × PB = PT².

How do you find the length of a tangent using a secant?

Find the whole secant PB = PA + AB, then use PT = √(PA × PB).

What is the intersecting chords theorem?

If two chords AB and CD meet at a point P inside a circle, then PA × PB = PC × PD.

Is the tangent–secant theorem in the Class 10 syllabus?

No. It is not part of the current CBSE Class 10 syllabus; it is enrichment material.