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Problem 170

Geometry Difficulty 1.2 Multiple choice CEMC Fermat · Canada · 2022

In the diagram, points $P(2,6)$ , $Q(2,2)$ and $R(8,5)$ form a triangle.

The area of $\triangle PQR$ is

Pick one

Official solution

The given triangle can be considered to have base $PQ$ (which is vertical along the line
$x=2$ ) and perpendicular height which
runs from $R$ horizontally to $PQ$ .

The line segment joining $P(2,6)$ to
$Q(2,2)$ has length 4.

Point $R(8,5)$ is 6 units to the
right of the line with equation $x = 2$ .

Thus, $\triangle PQR$ has area $\frac{1}{2}\cdot 4 \cdot 6 = 12$ .

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