$PT$ is a tangent to a circle at $T$, and $PAB$ is a secant. If $PT=6$ and $PA=4$, find $AB$.

Answer: 5 cm. $PT^2 = PA \times PB \Rightarrow 36 = 4 \times PB \Rightarrow PB = 9$. Since $PB = PA + AB$, $AB = 9 - 4 = 5$ cm.

Two chords $AB$ and $CD$ intersect internally at $P$. If $PA=4, PB=6, PC=3$, find $PD$.

Answer: 8 cm. Intersecting Chord Theorem: $PA \times PB = PC \times PD$. $24 = 3 \times PD \Rightarrow PD = 8$.

Geometry: Circles (Chords & Tangents)

Expert Answer & Key Takeaways

Master circle theorems including Tangent-Radius orthogonality, Alternate Segment Theorem, and Tangent-Secant power rules.

Model 1: Tangents & Radii

90° Rule: Radius is always perpendicular to the Tangent at the point of contact.
External Point: Two tangents drawn from an external point are equal ( $PA=PB$ ).

Model 2: The Golden Formula (Tangent-Secant)

PT^2 = PA \times PB

Where $PT$ is the tangent and $PAB$ is the secant line intersecting the circle at A and B.

Model 3: Alternate Segment Theorem

The angle between a tangent and a chord is equal to the angle in the alternate segment of the circle. This is a common 'hidden' property in SSC CGL problems.

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Geometry: Circles (Chords & Tangents)

Expert Answer & Key Takeaways

Model 1: Tangents & Radii

Model 2: The Golden Formula (Tangent-Secant)

Model 3: Alternate Segment Theorem

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