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TMUA 2022 · Paper 2 · Question 19 of 20

TMUA 2022 Paper 2 Question 19

Coordinate geometry — Cyclic polygons · when equal areas force regularity. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 2Coordinate geometryCyclic polygons · when equal areas force regularity5 optionshard
A polygon has n vertices, where n ≥ 3. It has the following properties:
  • Every vertex of the polygon lies on the circumference of a circle C.
  • The centre of the circle C is inside the polygon.
  • The radii from the centre of the circle C to the vertices of the polygon cut the polygon into n triangles of equal area.

For which values of n are these properties sufficient to deduce that the polygon is regular?

  1. Ano values of n
  2. Bn = 3 only
  3. Cn = 3 and n = 4 only
  4. Dn = 3 and n ≥ 5 only
  5. Eall values of n
Show the answer and worked solution
answer · B
  1. Ano values of n
  2. Bn = 3 only
  3. Cn = 3 and n = 4 only
  4. Dn = 3 and n ≥ 5 only
  5. Eall values of n
Each triangle has two sides equal to the radius r and area 12 r2 sinθi, so equal areas mean all sinθi agree, with ∑θi = 2π and every θi ∈ (0, π). Each angle is therefore θ or π − θ. For n = 3 neither mixed case can sum to 2π without pushing an angle to 0 or π, so all three are equal and the triangle is equilateral. For n = 4, taking two angles θ and two π − θ sums to 2π for any θ — that is a non-square rectangle. Similar mixtures exist for every larger n, so only n = 3 works.