Library / /393 of 428

, 2020

Geometry Difficulty 3.5 AMC 10/12 Find the answer Canada

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

Solution

For $BD$ to be a line of symmetry of $ABCD$ , the squares labelled $P$ and $S$ should be shaded. [[IMAGE0]] 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 $S$ ).

Figure for this problem

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

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