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Problem 48

Geometry Difficulty 1.0 Find the answer CEMC Fermat · Canada · 2020

The points $O(0, 0)$ , $P(0,3)$ , $Q$ , and $R(5,0)$ form a rectangle, as shown.

The coordinates of $Q$ are

$(5,5)$
$(5,3)$
$(3,3)$
$(2.5,1.5)$
$(0,5)$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solutions — 2

Solution 1

Since $OPQR$ is a rectangle with two sides on the axes, then its sides are horizontal and vertical.

Since $PQ$ is horizontal, the $y$ -coordinate of $Q$ is the same as the $y$ -coordinate of $P$ , which is 3.

Since $QR$ is vertical, the $x$ -coordinate of $Q$ is the same as the $x$ -coordinate of $R$ , which is 5.

Therefore, the coordinates of $Q$ are $(5,3)$ .

Solution 2

Since $OPQR$ is a rectangle with two sides on the axes, then its sides are horizontal and vertical.
Since $PQ$ is horizontal, the $y$ -coordinate of $Q$ is the same as the $y$ -coordinate of $P$ , which is 3.
Since $QR$ is vertical, the $x$ -coordinate of $Q$ is the same as the $x$ -coordinate of $R$ , which is 5.
Therefore, the coordinates of $Q$ are $(5,3)$ .

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