Mason wants to play with Maliyah's Doll House, but first he needs to stop at the clubhouse. If all three stops are in the shape of a triangle, which of the following distances would NOT be an option?

Answers :
5 6 9 city blocks
6 8 14
7 9 10
8 15 17

Here is their argument. given the obtuse angle x, we make a quadrilateral abcd with ∠dab = x, and ∠abc = 90◦, and ad = bc. say the perpendicular bisector to dc meets the perpendicular bisector to ab at p. then pa = pb and pc = pd. so the triangles pad and pbc have equal sides and are congruent. thus ∠pad = ∠pbc. but pab is isosceles, hence ∠pab = ∠pba. subtracting, gives x = ∠pad−∠pab = ∠pbc −∠pba = 90◦. this is a preposterous conclusion – just where is the mistake in the "proof" and why does the argument break down there?

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