Geometry - IOQM Study Guide
Geometry on IOQM is synthetic and ratio-driven: angle chasing, cyclic quadrilaterals, similar triangles, circle theorems, and area comparisons. A well-chosen auxiliary line or a correct ratio usually beats coordinate computation.
Core rules and concepts
- Angles in the same segment are equal; a quadrilateral is cyclic iff opposite angles are supplementary.
- Similar triangles: AA is enough; areas scale as the square of the linear scale factor.
- Area ratios: equal altitudes give bases ratio; equal bases give altitudes ratio.
- Power of a point: for intersecting chords PA·PB=PC·PD; tangent squared equals secant product.
- Use sine rule, cosine rule, and the angle bisector theorem when ratios are given.
Worked example
In triangle ABC, D lies on BC with BD:DC=2:3, and E is the midpoint of AD. Find [BEC]/[ABC]. Since BD:DC=2:3, [ABD]=2/5[ABC] and [ACD]=3/5[ABC]. E is the midpoint of AD, so triangles BDE and ABD share the altitude from B to AD; hence [BDE]=1/2[ABD]=1/5[ABC]. Similarly [CDE]=1/2[ACD]=3/10[ABC]. Now [BEC]=[BDE]+[CDE]=1/5+3/10=1/2 of [ABC]. If the answer is requested as a percentage, enter 50.
Traps that cost marks
- Assuming a point lies where the diagram suggests without proof.
- Using SSA as a congruence criterion.
- Confusing “opposite angles equal” with the cyclic condition.
- Applying power of a point with the wrong sign for an external point.
- Using the angle bisector theorem on the wrong pair of segments.