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Problem 589

AMC 10/12, early questions

Geometry Difficulty 3.5 Multiple choice CEMC Gauss (Grade 8) · Canada · 2020

In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}.$ Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$ .

If $\angle PUQ=\angle RVT =y^{\circ}$ , the value of $y$ is

Pick one

Official solution

For $BD$ to be a line of symmetry of $ABCD$ , the squares labelled $P$ and $S$ should be shaded.

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To see why this is, consider that the reflection in $BD$ of vertex $A$ is vertex $C$ , and so the reflection in $BD$ of side $DA$ is side $DC$ .

Further, the reflection in $BD$ of the first column of square $ABCD$ is the first row of square $ABCD$ .

Thus, the reflection in $BD$ of the shaded square in row 3, column 1 is the square in row 1, column 3 (the square labelled $P$ ).

More generally, the reflection in $BD$ of the square in row $r$ , column $c$ is the square in row $c$ , column $r$ .
Thus, the reflection in $BD$ of the shaded square in row 5, column 2 is the square in row 2, column 5 (the square labelled $S$ ).

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