Geometry: circle theorems, similarity and constructions
Geometry (WBBSE Madhyamik)
Geometry is the proof-heavy part of the paper: circle theorems, similarity and Pythagoras, plus constructions of the circumcircle, incircle and tangents. Marks in a proof go to the chain of reasoning, so every statement needs a reason - a named theorem, a given fact, or a construction.
Core rules
- The angle at the centre is twice the angle at the remaining circumference standing on the same arc.
- Angles in the same segment are equal; the angle in a semicircle is 90°.
- In a cyclic quadrilateral, opposite angles are supplementary and an exterior angle equals the interior opposite angle.
- Tangent properties: the tangent is perpendicular to the radius at the point of contact; tangents from an external point are equal; tangent² = product of the two secant segments from that point.
- Basic Proportionality (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio, and its converse also holds.
- Pythagoras: the square on the hypotenuse equals the sum of the squares on the other two sides; the converse detects a right angle.
Worked example
In cyclic quadrilateral ABCD, ∠A = 70°. Then ∠C = 180° − 70° = 110°, because opposite angles of a cyclic quadrilateral are supplementary. If AB is produced to E, the exterior angle ∠CBE equals the interior opposite angle ∠ADC.
Traps
- Quoting a theorem without naming it, or naming the wrong one.
- Assuming a quadrilateral is cyclic because it looks cyclic; cyclicity must be given or proved, for instance by equal angles on the same side of a line.
- In constructions, drawing freehand instead of showing the compass arcs that are the evidence of the method.
- Confusing the incircle (angle bisectors) with the circumcircle (perpendicular bisectors).