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Let $O$ be the origin. Points $P$ and $Q$ lie in the first quadrant. The slope of line segment $\overline{OP}$ is $4,$ and the slope of line segment $\overline{OQ}$ is $5.$ If $OP = OQ,$ then compute the slope of line segment $\overline{PQ}.$

 

Note: The point $(x,y)$ lies in the first quadrant if both $x$ and $y$ are positive.

 Apr 15, 2024
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Since O is the origin (0,0), the coordinates of P are of the form (a,4a) for some positive value a. Similarly, the coordinates of Q are of the form (b,5b) for some positive value b. We are given that OP=OQ, so a2+(4a)2​=b2+(5b)2​, which simplifies to a=b.

 

Therefore, the coordinates of Q are (a,5a), which means the slope of PQ​ is [\frac{(5a) - (4a)}{a - (0)} = \boxed{1}.]

 Apr 15, 2024

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