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Let $J$ and $K$ be given points along a certain line. There are exactly two other distinct points $U$ and $V$ on this line such that $UJ/UK=7$ and $VJ/VK=7$. Let $W$ be the midpoint of $\overline{UV}$. Calculate the ratio $WJ/WK$.

 Jun 22, 2022
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Since UJ/UK = 7, let UJ = 7x and UK = x.  Also, JV/JK = 7, so let VJ = 7y and VK = y.  Then JK = 6y, so 8x = 6y so x = 3y/4.

 

UV = UK + KV = x + y = 7y/4.  W is the midpoint of UV, so WU = UV = 7y/8.  Then WJ = 6y + 7y/8 = 55y/8 and WK = WU - KU = 7y/8 - x = 7y/8 - 3y/4 = y/8

 

So WJ/WK = (55y/8)/(y//8) = 55.

 Jun 23, 2022

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