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Problem 588

Geometry Difficulty 2.7 Multiple choice CEMC Gauss (Grade 8) · Canada · 2015

A rectangle has side lengths 3 and 4. One of its vertices is at the point $(1, 2)$ . Which of the following could not be the coordinates of one of its other vertices?

Pick one

Official solution

Since the sum of the measures of the three angles in any triangle is $180\degree$ , then the sum of the measures of the two unknown angles in the given triangle is $180\degree-45\degree=135\degree$ . The measures of the two unknown angles are in the ratio $4:5$ , and so one of the two angle measures is $\frac{5}{4+5}=\frac59$ of the sum of the two angles, while the other angle measures $\frac{4}{4+5}=\frac49$ of the sum of the two angles. That is, the larger of the two unknown angles measures $\frac59\times135\degree=75\degree$ , and the smaller of the unknown angles measures $\frac49\times135\degree=60\degree$ . We may check that $60\degree+75\degree+45\degree=180\degree$ . The largest angle in the triangle measures $75\degree$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.