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Geometry

Master the art of shapes, angles, and dimensions with visual logic and expert theorems.

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Introduction: The Visual Science

Geometry is not about measuring; it's about Seeing. In competitive exams, 70% of questions can be solved by identifying the hidden shape or property without heavy calculation.

Weightage: High in SSC CGL, CHSL, and increasing in Banking Mains.

1. Lines & Angles: The Alphabet

When a Transversal intersects two Parallel Lines, look for these 3 Letters:

1. The 'Z' Shape (Alternate Angles)

Angles inside the 'Z' are always EQUAL.

2. The 'F' Shape (Corresponding Angles)

Angles in the same position (like steps of a ladder) are EQUAL.

3. The 'C' Shape (Co-interior Angles)

Angles inside the 'C' sum to 180°.

2. Triangles: The Power of 3

The 4 Centers (Don't confuse them!)

Centroid (G)

Formed by Medians.
Center of Mass.
Divides median in 2:1.

Incenter (I)

Formed by Angle Bisectors.
Center of Incircle.
Equidistant from sides.

Circumcenter (O)

Formed by Perpendicular Bisectors.
Center of Circumcircle.
Equidistant from vertices.

Orthocenter (H)

Formed by Altitudes (Heights).
Intersection of perpendiculars from vertices.

Similarity (The Photo-Resize Rule)

If $\Delta ABC \sim \Delta PQR$ , then sides are proportional, but Areas follow the Square Rule:

\frac{\text{Area}(\Delta ABC)}{\text{Area}(\Delta PQR)} = (\frac{AB}{PQ})^2

3. Circles: The Sniper Methods

Tangent-Radius Rule Radius is always Perpendicular (90°) to the Tangent at the point of contact. This creates Right Angled Triangles $\to$ Use Pythagoras!
The "Ice-Cream" Cone Two tangents drawn from an external point are always EQUAL in length ( $PA = PB$ ).
Alternate Segment Theorem (The Hidden Equal) The angle between a tangent and a chord is equal to the angle in the alternate segment. (Look for this in complex figures!).

The Golden Formula (Tangent-Secant)

PT^2 = PA \times PB

(Where $PT$ is tangent, $PAB$ is secant)

4. Polygons: The Formula Sheet

For a regular polygon with n sides:

Property	Formula
Sum of Interior Angles	$(n-2) \times 180^\circ$
Each Interior Angle	$\frac{(n-2) \times 180}{n}$
Sum of Exterior Angles	$360^\circ$ (Always!)
Each Exterior Angle	$\frac{360}{n}$
Diagonals	$\frac{n(n-3)}{2}$

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