## CSMC 2023 Part B - Question 1, CEMC UWaterloo

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#### Difficulty: 6

#### This problem is tagged with csmc, csmc23, highschool.

**(Canadian Senior Mathematics Contest 2023, Part B, Question 1, CEMC - UWaterloo)**

In the diagram, $ABCD$ is a trapezoid with $\angle DAB = \angle ADC = 90^{\circ}$. Also, $AB = 7$, $DC = 17$, and $AD = 10$. Point $P$ is on $AD$ and point $Q$ is on $BC$ so that $PQ$ is parallel to $AB$ and to $DC$. Point $F$ is on $DC$ so that $BF$ is perpendicular to $DC$. $BF$ intersects $PQ$ at $T$.

a) Determine the area of trapezoid $ABCD$.

b) Determine the measure of $\angle BQP$.

c) If $PQ=x$, determine the length of $AP$ in terms of $x$.

d) Determine the length of $PQ$ for which the areas of trapezoids $ABQP$ and $PQCD$ are equal.

*Answer Submission Note(s)*

Your answer must contain an integer for the first part, a single integer (no degree symbol) for the second part, a simple equation (with no spaces) for the third part, and a single integer for the final part. Separate part answers with commas (no spaces).

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