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Problem 829

AMC 10/12, early questions

Geometry Difficulty 3.7 Multiple choice South African Mathematics Olympiad · South Africa

The number of squares that have $(-1; -1)$ as a vertex and at least one of the coordinate axes as an axis of symmetry is

Pick one

Official solution

Answer E.
If we reflect $P(-1; -1)$ in the $x$ -axis to map to $Q(-1; 1)$ we obtain 3 squares with the $x$ -axis as a line of symmetry, as shown in the diagram. Two of the cases have $PQ$ as an edge/side and in one case $PQ$ is a diagonal of the square. Similarly, if we reflect $P$ in the $y$ -axis to $R$ , then there are three possible squares. However, one of these is the same as before, so in total there are exactly five squares possible.

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