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?

aboslutelydestroying Apr 7, 2024

#1**+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!

aboslutelydestroying Apr 7, 2024

#2**+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

CPhill Apr 7, 2024