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Determine $k$ so that the points $A(7,5), B(-1,2)$, and $C(k, 0)$ are the vertices of a right triangle with right angle at B.
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## 1 Answer

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The condition for $\triangle A B C$ to have a right angle at $B$ is that lines $\overrightarrow{A B}$ and $\overleftrightarrow{B C}$ are perpendicular, which holds when and only when the product of their slopes is $-1$, that is $\left(\frac{3}{8}\right)[-2 /(1+k)]=-1$. This is equivalent to $6=8(1+k), \quad$ or $8 k=-2$, or $k=-\frac{1}{4}$.
by Platinum (164,234 points)

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