Circle Theorems: The Invariable Scoring Questions
Circle properties account for over 50% of all geometry questions asked by the Staff Selection Commission. The following three theorem models appear in almost every examination session:
- Direct Common Tangent: Length = √[d² - (r₁ - r₂)²], where d is distance between centers.
- Transverse Common Tangent: Length = √[d² - (r₁ + r₂)²].
- Intersecting Chords Theorem: When chords AB and CD intersect at P, PA × PB = PC × PD (applies inside or outside circle).
- Tangent-Secant Theorem: If PT is tangent and PAB is secant, PT² = PA × PB.
Key Strategy Takeaway
If you memorize Direct and Transverse Common Tangent formulas, you guarantee 2 to 4 marks in every SSC CGL exam.
Triangle Centers & Radii Relationships
Understand the clear distinction between Incenter, Circumcenter, Centroid, and Orthocenter:
- Inradius (r) = Area (Δ) / Semi-perimeter (s). For right-angled triangle: r = (P + B - H) / 2.
- Circumradius (R) = (a × b × c) / (4Δ). For right-angled triangle: R = Hypotenuse / 2.
- Centroid divides each median in a 2:1 ratio from vertex to midpoint.
- Euler Line: Orthocenter, Centroid, and Circumcenter are collinear; Centroid divides H-O line in 2:1.
3D Mensuration: Cones, Frustums & Melting Solids
Volume conservation during recasting (e.g., spheres melted into cylinders) is the most common 3D problem. Remember that when multiple identical spheres are formed from a larger sphere of radius R, n × r³ = R³.