Geometry (Triangles, Quadrilaterals, Circles)
What this domain tests
Geometry tests similar triangles, the Basic Proportionality Theorem, Pythagoras' theorem, and circle theorems about tangents and cyclic quadrilaterals. Proofs and short calculations both appear.
Core rules
- Similar triangles: corresponding angles are equal and corresponding sides are proportional. Valid criteria are AAA, SAS and SSS.
- Basic Proportionality Theorem (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio.
- Areas of similar triangles are in the ratio of the squares of corresponding sides.
- Pythagoras: in a right triangle, hypotenuse² = sum of the squares of the other two sides; the converse also holds.
- A tangent is perpendicular to the radius at the point of contact. Tangents from an external point are equal in length.
- The angle in a semicircle is 90°. Opposite angles of a cyclic quadrilateral are supplementary.
Worked example
A tangent is drawn from an external point P to a circle of radius 5 cm, with OP = 13 cm. Find the tangent length. Let T be the point of contact. The radius OT is perpendicular to PT, so triangle OTP is right-angled at T. PT² = OP² − OT² = 13² − 5² = 169 − 25 = 144. PT = 12 cm. The right angle at the point of contact is the key step.
Traps
- Treating SSA as a similarity criterion.
- Matching corresponding vertices in the wrong order.
- Using the side ratio instead of its square for areas.
- In tangent problems, forgetting that the radius meets the tangent at 90°.
- Confusing opposite angles with adjacent angles in a cyclic quadrilateral.