Chapter 8 · Kerala SSLC Class 10 Maths
Tangents
Two ways to learn — a crisp formula reference, or a story that makes the geometry click.
A story that builds each theorem naturally.
How story mode works
Each scene is a real situation that naturally reveals a tangent theorem. Read the story, then tap "The Maths" to see the formal statement.
A Ball Rolling Against a Wall — Tangent ⊥ Radius
Picture a perfectly round football resting against a flat wall. The wall touches the ball at exactly one point — the point of contact. Now draw an imaginary line from the centre of the ball to that contact point.
The wall must be perpendicular to that radius. If it were tilted even slightly, the ball would either go through the wall or not touch it. A tangent can only exist when it is at exactly 90° to the radius at the contact point.
This is not just intuition — it is a theorem. The radius OT is always perpendicular to the tangent PT at the point of contact T.
Two Rubber Bands from a Point — Equal Tangent Lengths
Stretch two rubber bands from the same pin on a table, each wrapped around a circular tin. No matter how you place the tin, both rubber bands that leave from the same pin and just touch the tin are always the same length.
Is that always true? It feels like it could be slightly different each time.
Always. And we can prove it. Both triangles — pin to centre to contact point A, and pin to centre to contact point B — are congruent.
In △OAP and △OBP:
OA = OB (radii, same circle)
OP = OP (common side)
∠OAP = ∠OBP = 90° (tangent ⊥ radius)
∴ △OAP ≅ △OBP (RHS)
∴ PA = PB
The two tangent lengths from any external point are always equal. This fact is used constantly in SSLC problems.
The Sliding Ruler — Tangent-Chord Angle Theorem
Place a ruler as a tangent touching a circle at point T. Now draw a chord TC inside the circle. The angle the chord makes with the tangent on one side equals the angle that TC subtends in the opposite (alternate) segment of the circle.
So the angle at the edge — between the tangent and the chord — equals an angle sitting deep inside the circle on the other side?
Exactly. Slide the ruler (change the tangent direction) and both angles change together, always staying equal. This is the alternate segment theorem.
If the angle in the alternate segment is 65°, then the angle between the tangent and the chord is also 65°.
The big picture
Every tangent question uses one of three facts. Know which fact to reach for and the rest is substitution.
Need to find tangent length?
PT = √(OP² − r²) (Pythagoras on OT ⊥ PT)
Two tangents from the same point?
PA = PB (equal tangent lengths)
Angle between tangent and chord?
∠(tangent,chord) = ∠ in alternate segment
Angle between two tangents?
∠APB + ∠AOB = 180°
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