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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 30, 2024
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Suppose P = (x, y). Since the slope of OP is 4, by slope formula, we have y0x0=4, which means y=4x.

 

So, P = (x, 4x) for some x. By the same argument, Q = (w, 5w) for some w. 

 

Now, we use the condition OP=OQ to find a connection between w and x. Using distance formula to calculate OP and OQ, we have

x2+(4x)2=w2+(5w)2

17x2=26w2

x=2617w

 

Now, the slope of PQ is 5w4xwx=5w42617ww2617w=5174261726

 

Rationalizing gives mPQ=(517426)(17+26)1726=19+4429=194429.

 

If you just need an approximate value, 1944290.2249.

 Apr 30, 2024

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