Geometry & Triangles (Theorems, Similarity, Circles)

Geometry & Triangles (Theorems, Similarity, Circles)

Concept

Geometry in SSC CGL Quant is visual and theorem-based. You need to recall and apply specific theorems quickly. The most tested concepts are: Pythagoras theorem and its applications (3-4-5, 5-12-13, 8-15-17 triangles), similarity criteria (AA, SAS, SSS), and circle theorems (angle at centre, chord equalities, tangent properties).

Triangle Similarity: Two triangles are similar if: (AA) two angles equal → third automatically equal; (SAS) ratio of two sides equal with the included angle equal; (SSS) all three sides proportional. When triangles are similar, corresponding sides are in the same ratio, and areas are in the ratio of squares of corresponding sides.

Important Theorems:

• Apollonius’s Theorem: In any triangle, median to side a: AB² + AC² = 2(AD² + BD²) where D is midpoint of BC.
• Angle Bisector Theorem: The internal angle bisector divides the opposite side in the ratio of the adjacent sides: BD/DC = AB/AC.
• Mid-Point Theorem: Line joining midpoints of two sides is parallel to the third side and half its length.
• Pythagoras triplet generation: If m > n, sides = m²−n², 2mn, m²+n² gives a primitive triplet.

Circle Theorems:

• Equal chords subtend equal angles at the centre.
• Perpendicular from centre to a chord bisects the chord.
• Tangent at any point is perpendicular to the radius at that point.
• Two tangents drawn from an external point are equal in length.

Key Points

• Area ratio of similar triangles = (side ratio)². If sides are in ratio 2:3, areas are in ratio 4:9.
• For any triangle: area = √[s(s−a)(s−b)(s−c)] where s = semi-perimeter (Heron’s formula).
• In a circle, angle at centre = 2 × angle at circumference (subtended by same arc).
• Opposite angles of a cyclic quadrilateral sum to 180°.
• If two circles touch externally: distance between centres = sum of radii. Internally: difference of radii.

Worked Example

Q: In a triangle ABC, AD is the median to BC. If AB = 10, AC = 8, BC = 12, find AD. Approach: Apollonius: AB² + AC² = 2(AD² + (BC/2)²). 100 + 64 = 2(AD² + 36). 164 = 2AD² + 72 → 2AD² = 92 → AD² = 46 → AD = √46. Answer: √46

SSC Pattern / Tips

• Pythagorean triplets save time — know 3-4-5, 5-12-13, 8-15-17, 7-24-25, 15-20-25.
• For similar triangles, always identify which sides correspond to which before writing ratios.
• In cyclic quadrilateral problems, look for pairs of opposite angles summing to 180°.
• For tangents from an external point, always draw the radii to both points of tangency — they form right angles with the tangents.