+0  
 
0
7
2
avatar+214 

The points (1, 7), (13, 16) and (7,k), where k is an integer, are vertices of a triangle. What is the sum of the values of kfor which the area of the triangle is a minimum?

 #1
avatar+214 
+1

CPhill, after answering this question, are you able to check on the last problem, becuase I don't think your answer is correct. Thank you!smiley

 #2
avatar+128794 
+1

Slope between the frist two points

[16-7]/[13-1]= 9/12  = 3/4

 

Equation of line through these two points

 

y = (3/4) (x -1) + 7           put into  standard form

 

4y = 3(x -1) + 28

 

4y = 3x - 3 + 28

 

3x - 4y + 25  = 0

 

Using  the partial  formula for the distance  between a  point and  a line,  we want to minimize this

 

l 3(7) - 4(k) + 25 l  = 0 

 

l-4k + 46 l  = 0

 

-4k   = -46

 

k = 46/4  = 11.5

 

The integer  values  above and  below this that are closest to 11.5  are  k = 11  and  k =12

 

Their sum = 23

 

Here's a graph

 

 

cool cool cool 

 Apr 7, 2024

1 Online Users