The points (1, 7), (13, 16) and (6, k), where k is an integer, are vertices of a non-degenerate triangle. What is the sum of the values of k for which the area of the triangle is a minimum?
+128448 There is only one integer satisfying k
The slope between (1,7) and (13,16) = 9/12 = 3/4
The equation of the line containing this segment is
y = (3/4) (x - 1) + 7 ⇒ 3x - 4y + 25 = 0
The distance shortest distance between this line and ( 6 , k) is given by
l 3(6) - 4k + 25 l / sqrt ( 3^2 + 4^2)
l 43 - 4k l / 5 or l 4k - 43 l / 5
Note that the distance must be poisitive
So
4k - 43 > 0
4k > 43
k > 43 / 4
So....since k must be an integer. > 43/4...then....
Ceiling (43/4) = 11 = k
