Triangles and Geometric Constructions
Similarity criteria, the basic proportionality theorem, the area ratio, and Pythagoras carry most of the theory marks; the construction question demands accurate arcs plus written steps. Both halves must be answered.
Core rules
- Similarity: AA (two equal angles), SSS (all three pairs of sides proportional), SAS (two sides proportional with the included angle equal).
- Basic Proportionality Theorem: in triangle ABC, if DE ∥ BC with D on AB and E on AC, then AD/DB = AE/EC, and also AD/AB = AE/AC.
- Areas of similar triangles are in the ratio of the squares of corresponding sides.
- Pythagoras: in a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides; the converse justifies a right angle.
- Construction of division in ratio m : n: draw a ray from A, mark m + n equal arcs with a compass, join the last to B, then draw a parallel through the m-th point.
- Tangent construction from external point P: bisect OP, draw a circle with that midpoint as centre through P; its intersections with the given circle are the contact points.
Worked example
In triangle ABC, DE ∥ BC with AD = 3 cm, DB = 5 cm and AE = 4.5 cm. Find EC. By the Basic Proportionality Theorem, AD/DB = AE/EC, so 3/5 = 4.5/EC, giving EC = (5 × 4.5)/3 = 7.5 cm.
Traps
- Writing AD/AB = AE/EC, which mixes a part with a whole.
- Using the side ratio, not its square, for areas of similar triangles.
- Erasing construction arcs; they are evidence of the method.
- Omitting the steps of construction, which carry separate marks.
- Treating similar triangles as congruent and equating sides.