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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)

 Feb 5, 2021
 #1
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Not totally sure about this one, geoNewbie, but here's  what I get

 

We  have  a known (fixed) side that is  15  units in length   { (1,7)  to  (13,16) }   (5,k)

 

So   one of these must be true

 

sqrt [ (1-5)^2  +(7-k)^2]  + sqrt [ ( 13-5)^2  + (16-k)^2] > 15

 

sqrt  [ ( 13 - 5)^2  + ( 16-k)^2 ] + 15  > sqrt [ ( 1-5)^2  + ( 7 - k)^2 ] 

 

15  + sqrt [ ( 1-5)^2  + ( 7 - k)^2 ]  >  sqrt  [ ( 13 - 5)^2  + ( 16-k)^2 ]

 

These are a little  messy to solve,  but  using Wolframalpha, I only get a definite  solution for the  first

 

k < 10    or  k > 10

 

So   the points   (5,9)   and (5,11)   seem to  be  the answers

 

Maybe someone else has a  better solution  ????

 

cool cool cool

 Feb 5, 2021

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