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, 2012

Geometry Difficulty 1.5 Junior Find the answer Canada

In the diagram, the horizontal distance between adjacent dots in the same row is 1. Also, the vertical distance between adjacent dots in the same column is 1.

What is the perimeter of quadrilateral $PQRS$ ?

Your answer

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

Solution

The perimeter of quadrilateral $PQRS$ equals $PQ + QR + RS + SP$ .

Since the dots are spaced 1 unit apart horizontally and vertically, then $PQ = 4$ , $QR=4$ , and $PS=1$ .

Thus, the perimeter equals $4 + 4 + RS + 1$ which equals $RS + 9$ .

We need to determine the length of $RS$ .

If we draw a horizontal line from $S$ to point $T$ on $QR$ , we create a right-angled triangle $STR$ with $ST = 4$ and $TR=3$ .

By the Pythagorean Theorem, $RS^2 = ST^2 + TR^2 = 4^2 + 3^2 = 25$ .

Since $RS>0$ , then $RS = \sqrt{25}=5$ .

Thus, the perimeter of quadrilateral $PQRS$ is $5+9=14$ .

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