Grade 11 Euclidean Geometry • Theorem 4

The Tan-Chord Theorem

The Intuition (Plain English)

When a straight line (a tangent) just grazes the edge of a circle, and a chord meets it at that exact point of contact, the angle trapped between the tangent and the chord "jumps" across the circle. It will be exactly equal to the angle inside the triangle on the opposite side of the chord.

The Golden Rule: Look for the "V" or "Z" shape resting on top of a tangent line. The angle outside the triangle equals the angle inside the far corner.

The Formal Proof

This is one of the 6 examinable proofs. Examiners specifically want to see you construct a diameter to force a 90° angle.

📐 [Insert GeoGebra Diagram] Circle O, tangent PQ at B. Chord AB. Triangle ABC inside circle.

Given: Circle with centre O. Tangent PQ touches the circle at B. AB is a chord and ΔABC is inscribed in the circle.

R.T.P: ∠ABP = ∠C

Construction: Draw diameter BOD and join AD.

Statement	Reason
1. ∠DBP = 90°	tan ⊥ radius (or diameter)
2. ∴ ∠DBA + ∠ABP = 90°	adjacent angles
3. ∠DAB = 90°	∠in semi circle
4. ∠D + ∠DBA = 90°	sum of ∠s in ΔDAB is 180°
5. ∴ ∠ABP = ∠D	both equal 90° - ∠DBA
6. But ∠D = ∠C	∠s in the same seg (subtended by arc AB)
7. ∴ ∠ABP = ∠C	proven

How to apply this in an exam (The "Hidden Parallel" Trick)

Examiners love combining the Tan-Chord theorem with parallel lines. This forces you to use alternate or corresponding angles alongside circle geometry.

Typical Exam Question:
PQ is a tangent to a circle at B. AC is a chord such that AC ∥ PQ. Prove that ΔABC is an isosceles triangle.

Step-by-Step Solution:

Let the angle between the tangent PQ and chord AB be ∠ABP.
∠C = ∠ABP (tan chord theorem)
Because AC ∥ PQ, we have an alternate angle:
∠A = ∠ABP (alt ∠s; AC ∥ PQ)
Since ∠C = ∠ABP and ∠A = ∠ABP, it must be true that ∠A = ∠C.
Therefore, AB = BC (sides opp equal ∠s), which means ΔABC is an isosceles triangle.

⚠️ Common Exam Traps

Assuming a line is a tangent: Just because a line looks like it only touches the circle once, you cannot use the Tan-Chord theorem unless the question specifically states "PQ is a tangent".
Jumping to the wrong segment: The angle between the tangent and chord A is equal to the angle subtended by chord A on the *opposite* side of the circle. Trace your finger from the chord across the circle to the far corner.