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tangent-secant theorem — Definition, Formula & Examples

The tangent-secant theorem says that when a tangent and a secant are drawn from the same external point to a circle, the tangent length squared equals the product of the secant's external segment and its entire length.

If a tangent segment of length tt and a secant segment with external part aa and whole length bb are drawn from a common external point to a circle, then t2=a⋅bt^2 = a \cdot b. This is a special case of the power-of-a-point theorem.

Key Formula

t2=a⋅bt^2 = a \cdot b
Where:
  • tt = Length of the tangent segment from the external point to the point of tangency
  • aa = Length of the external part of the secant (external point to near intersection)
  • bb = Total length of the secant (external point to far intersection)

How It Works

Identify the external point outside the circle. Measure the tangent segment from that point to the point of tangency — call its length tt. For the secant, measure the distance from the external point to the near intersection with the circle (aa) and to the far intersection (bb). The theorem guarantees t2=a⋅bt^2 = a \cdot b. You can use this relationship to find any one unknown when the other two lengths are known.

Worked Example

Problem: From a point outside a circle, a tangent of length 6 and a secant are drawn. The secant's external segment is 4. Find the total length of the secant.
Write the theorem: Apply the tangent-secant relationship with t=6t = 6 and a=4a = 4.
t2=a⋅bt^2 = a \cdot b
Substitute and solve: Plug in the known values and solve for bb.
62=4⋅b  ⟹  36=4b  ⟹  b=96^2 = 4 \cdot b \implies 36 = 4b \implies b = 9
Answer: The total length of the secant is 9, so the chord portion inside the circle is 9−4=59 - 4 = 5.

Why It Matters

This theorem appears frequently in high-school geometry proofs and standardized tests (SAT, ACT) whenever a figure mixes tangent and secant lines. It is also essential in engineering and surveying when you need to calculate inaccessible distances using measurements taken from an external point.

Common Mistakes

Mistake: Using only the internal chord portion of the secant instead of the full secant length for bb.
Correction: The variable bb must be the entire secant length from the external point through the circle to the far intersection, not just the chord inside the circle. Always add the external segment to the chord: b=a+chord lengthb = a + \text{chord length}.

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