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# coordinates

0
89
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(1,7), (13,-16) and (5,k) are the vertices of a triangle. What is the sum of all possible values of "k" for which the area of the triangle is minimum? (k can only be an integer)

Jun 11, 2022

#1
+2448
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The equation of the line in point-slope form is: $$y - 7 =-\frac{23}{12}\left(x-1\right)$$

Now, plugging in $$x = 5$$, we find $$y = -{2 \over 3}$$.

This means when $$k = -{2 \over 3}$$, the point form a straight line, so we naturally want k to be the closest integer to $$-{2 \over 3}$$, which is $$\color{brown}\boxed{-1}$$

Jun 11, 2022