Geometry: Similar Triangles, Circle Theorems and Constructions
Geometry questions come from the Basic Proportionality Theorem and similarity criteria, area ratios for similar triangles, tangent and chord circle theorems, and ruler–compass constructions. Proofs carry step marks, so every statement needs a reason.
Triangles and similarity
- BPT (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio; the converse is also a theorem.
- Similarity criteria: AA (two angles equal), SSS (all three side pairs in the same ratio), SAS (two sides proportional with the included angle equal).
- Areas of similar triangles are in the ratio of the squares of corresponding sides; perimeters are in the ratio of the sides.
- Pythagoras: in a right triangle, hypotenuse² = sum of squares of the legs; the converse identifies right angles.
Circles
- A tangent is perpendicular to the radius at the point of contact.
- Tangents drawn from an external point are equal in length, and the line from the centre bisects the angle between them.
- Angle in a semicircle is 90°; the angle between a tangent and a chord equals the angle in the alternate segment.
Constructions
- To divide a segment in m:n, draw a ray, mark m + n equal arcs, join the last point to the other end and draw a parallel through the m-th arc.
- For tangents from an external point P, construct the circle on OP as diameter; its intersections with the given circle are the points of contact.
Worked example
In △ABC, DE ∥ BC with AD = 2 cm, DB = 3 cm, AE = 4 cm. Find AC. By BPT, AD/DB = AE/EC, so 2/3 = 4/EC, giving EC = 6 cm and AC = AE + EC = 10 cm.
Traps that cost marks
- Ignoring the order in △ABC ~ △DEF, which fixes which sides correspond.
- Using the side ratio instead of its square for areas.
- Assuming a tangent passes through the centre.
- Marking unequal arcs in a construction, which destroys the required ratio.