Euclidean Geometry: Triangles, Circles and Constructions
Euclidean Geometry: Triangles, Circles and Constructions tests theorem-based reasoning and accurate drawing. You must state the theorem used, match corresponding parts, and justify constructions with geometry, not appearance.
Core rules
- Basic proportionality theorem: a line parallel to one side of a triangle divides the other two sides proportionally; the converse also holds.
- Similar triangles: AA, SAS similarity and SSS similarity; corresponding sides are proportional and corresponding angles equal.
- Pythagoras theorem and its converse; in a right triangle, the square on the hypotenuse equals the sum of squares on the other two sides.
- Circle: equal chords are equidistant from the centre; the perpendicular from the centre bisects a chord; angle in a semicircle is 90°; tangent is perpendicular to the radius at the point of contact.
- Construction: bisect an angle, draw perpendicular bisector, construct tangents, and divide a segment in a given ratio; retain arcs and label them.
Worked example
In triangle ABC, DE∥BC with AD=2 cm, DB=3 cm and AE=4 cm. By BPT, AD/DB=AE/EC, so 2/3=4/EC, giving EC=6 cm.
Traps
Do not assume a line is parallel because it looks so; prove it. In similar triangles, match vertices in the correct order. In circle problems, the tangent-radius right angle is at the point of contact, not at the centre. Constructions need arc evidence; a freehand line loses marks.