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${{\bullet\!\!\! -\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\bullet\!\!\! -\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\bullet}\atop O \;\quad\quad M\quad\quad\; P}$

Given that $OM = x + 8$, $MP = 2x - 6$, $OP = 44$, is $M$ the midpoint of $OP$?

2 Answers 2

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HINT $OM + MP = OP$ Move your mouse over the gray area for the complete solution.

Note that $OM + MP = OP$ and hence we get that $3x+2 = 44 \implies x =14$. Hence, $OM = 14 + 8 = 22 = \dfrac{44}2 = \dfrac{OP}2$. Hence, $M$ is indeed the midpoint.

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    @Michael x^2 + 2x = 8 \, \& \, 2x+4 \geq 0 \implies x=22012-06-28
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We have that $OM + MP = OP$ so $(x+8)+(2x-6)=3x+2=44$ $3x=42$ $x=14$ Therefore $OM=x+8=22$ and $MP=2\cdot 14-6=28-6=22$. Because the lengths of $OM$ and $MP$ are equal, $M$ is the midpoint of the line $OP$.