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TMUA 2016 · Paper 2 · Question 9 of 20

TMUA 2016 Paper 2 Question 9

Coordinate geometry — Congruent triangles · what equal area adds to the standard criteria. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 2Coordinate geometryCongruent triangles · what equal area adds to the standard criteria8 options
Triangles ABC and XYZ have the same area.

Which of these extra conditions, taken independently, would imply that they are congruent?

(1)  AB = XY and BC = YZ
(2)  AB = XY and ∠ ABC = ∠ XYZ
(3)  ∠ ABC = ∠ XYZ and ∠ BCA = ∠ YZX

Each option gives a verdict for condition (1), then (2), then (3).

  1. A(1) does not imply congruent; (2) does not imply congruent; (3) does not imply congruent
  2. B(1) does not imply congruent; (2) does not imply congruent; (3) implies congruent
  3. C(1) does not imply congruent; (2) implies congruent; (3) does not imply congruent
  4. D(1) does not imply congruent; (2) implies congruent; (3) implies congruent
  5. E(1) implies congruent; (2) does not imply congruent; (3) does not imply congruent
  6. F(1) implies congruent; (2) does not imply congruent; (3) implies congruent
  7. G(1) implies congruent; (2) implies congruent; (3) does not imply congruent
  8. H(1) implies congruent; (2) implies congruent; (3) implies congruent
Show the answer and worked solution
answer · D
  1. A(1) does not imply congruent; (2) does not imply congruent; (3) does not imply congruent
  2. B(1) does not imply congruent; (2) does not imply congruent; (3) implies congruent
  3. C(1) does not imply congruent; (2) implies congruent; (3) does not imply congruent
  4. D(1) does not imply congruent; (2) implies congruent; (3) implies congruent
  5. E(1) implies congruent; (2) does not imply congruent; (3) does not imply congruent
  6. F(1) implies congruent; (2) does not imply congruent; (3) implies congruent
  7. G(1) implies congruent; (2) implies congruent; (3) does not imply congruent
  8. H(1) implies congruent; (2) implies congruent; (3) implies congruent
Use area = 12 absin C throughout. For (1), equal areas with AB = XY and BC = YZ force sin B = sin Y, but that leaves Y = 180∘ − B open, so the triangles need not be congruent. For (2), AB = XY and ∠ B = ∠ Y make the areas 12 AB⋅ BCsin B and 12 XY⋅ YZ sin Y equal only if BC = YZ, and then SAS gives congruence. For (3), two equal angles make the triangles similar, and similar triangles with equal area have scale factor 1, so they are congruent.