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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?

 Apr 23, 2022
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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

 

cool cool cool

 Apr 23, 2022

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