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PEI Math 521B study materials
PEI Math 521B study resources

PEI Math 521B: Geometry

PEI Math 521B · Work through your exam, one topic at a time.

All guide sections

Geometry

What this domain tests

Geometry on PEI Math 521B emphasizes deductive argument: congruence and similarity, circle theorems, coordinate and vector methods, and transformations. Questions often present a configuration in words and ask for a proof or a computed length or angle. You are expected to justify each step by a named theorem rather than by appearance of a diagram.

Core rules and concepts

  • Congruence criteria: SSS, SAS, ASA, AAS, and HL for right triangles. SSA is not a congruence criterion in general.
  • Similarity: AA suffices for triangles; corresponding sides are proportional, and areas scale as the square of the ratio.
  • Circle theorems: the inscribed angle is half the central angle on the same arc; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; the tangent is perpendicular to the radius at the point of contact.
  • Power of a point: for a point P and a circle, the product of the secant segments is constant, and equals the square of the tangent length when P is outside.
  • Coordinates and vectors: the midpoint formula, distance formula, and dot product (zero for perpendicular vectors) convert synthetic claims into algebra.

Worked example

A cyclic quadrilateral ABCD has ∠ABC = 110°. Find ∠ADC, then find the angle subtended at the centre by arc AC not containing B, given that ∠ABC subtends the other arc. Since ABCD is cyclic, opposite angles are supplementary: ∠ADC = 180° − 110° = 70°. The inscribed angle ∠ABC = 110° intercepts arc ADC, so that arc measures 220°, and the remaining arc AC measures 360° − 220° = 140°. The central angle on that arc is therefore 140°. Each step cites a theorem: cyclic quadrilateral, inscribed angle equals half its intercepted arc, and the full circle is 360°.

Traps that cost marks

  • Using SSA to claim congruence, especially in ambiguous cases with two possible triangles.
  • Confusing the arc intercepted by an inscribed angle with the arc it "faces" on a sketch.
  • Applying power of a point with segments measured from different endpoints, so products are mismatched.
  • Computing an area ratio as the side ratio rather than its square in similarity problems.