Geometry and Mensuration
Syllabus Unit: Quantitative Ability (QA)
Why it matters
Chapter overview
CAT geometry rewards recognising the right theorem fast — angle properties in triangles, tangent-radius relationships in circles, area/perimeter formulas for polygons, and coordinate geometry that turns shapes into equations — over lengthy constructions, since every question must be solved with pen and paper alone under time pressure.
1Lines, Angles & Triangles
Every triangle carries three special centres that CAT tests by name and ratio — know all three cold.
Learning Objectives
- Apply the triangle angle sum property (180°) and the exterior angle theorem (exterior = sum of two remote interior angles).
- Distinguish the centroid, orthocentre and circumcentre by their construction and their signature ratios.
- Apply triangle inequality: the sum of any two sides must exceed the third side.
- Use similarity (AA, SAS, SSS criteria) to set up ratio equations between corresponding sides.
Key Terms
Triangle problems on CAT usually hinge on one of three special points — the **centroid**, **orthocentre**, or **circumcentre** — each defined by a different set of triangle lines, and each carrying its own signature property.
The three centres
- Centroid: meeting point of the three medians (vertex to midpoint of opposite side); divides each median 2:1 from the vertex.
- Orthocentre: meeting point of the three altitudes (perpendiculars from a vertex to the opposite side).
- Circumcentre: meeting point of the perpendicular bisectors of the sides; equidistant from all 3 vertices, and is the centre of the circle passing through them.
Similarity and the triangle inequality
Two triangles are similar if their corresponding angles are equal (AA is sufficient) — similarity gives proportional sides, which is how most CAT triangle ratio problems are actually solved.