Expert: Sherman D. - 10/22/2006

Question
a shopping mall has entrances located at A(4,10), B(5,7), and C(-7,3). An information booth is to be located at a point that is equidistant from each entrance. Determine the position of the booth.

i did this question, and got my two equations, and did substitution, but im not sure if the two coordinates are right.

Answer
(5,7) and (-7,3)
M = (-7 + 5)/2 , (3 + 7)/2
M = (-2/2), (10/2)
M = -1,5

(4,10) and (-7,3)
M = (-7 + 4)/2 , (10 + 3)/2
M = (-3/2) and (13/2)
M = -1.5,6.5

(4,10) and (5,7)
M = (5 + 4)/2 , (10 + 7)/2
M = (9/2),(17/2)
M = 4.5,8.5

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(4,10) and (-1,5)
m = (5 - 10)/(-1 - 4)
m = (-5)/(-5)
m = 1

(4,10), m = 1
10 = 1(4) + b
10 = 4 + b
b = 6

y = x + 6

(-7,3) and (4.5,8.5)
m = (8.5 - 3)/(4.5 + 7)
m = (5.5)/(11.5)
m = (11/2)/(23/2)
m = (11/23)

(-7,3), m = (11/23)
3 = (11/23)(-7) + b
3 = (-77/23) + b
69 = -77 + 23b
23b = 146
b = (146/23)

y = (11/23)x + (146/23)

(-1.5,6.5) and (5,7)
m = (7 - 6.5)/(5 + 1.5)
m = (1/2)/(13/2)
m = (1/13)

(5,7), m = (1/13)
7 = (1/13)(5) + b
7 = (5/13) + b
91 = 5 + 13b
86 = 13b
b = (86/13)

y = (1/13)x + (86/13)

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y = (11/23)x + (146/23)
y = (1/13)x + (86/13)
y = x + 6

(11/23)x + (146/23) = (1/13)x + (86/13)
(143/299)x + (1898/299) = (23/299)x + (1978/299)
(120/299)x = (80/299)
120x = 80
x = (2/3)

y = (11/23)(2/3) + (146/23)
y = (22/69) + (146/23)
y = (22/69) + (438/69)
y = (460/69)
y = (20/3)

(20/3) = (2/3) + 6
(20/3) = (20/3)

ANS : ((2/3),(20/3))

What i did was i used the vertex coordinate with with the coordinate of the midpoint of the opposite side, then i set them equal to each other so that the distance between each vertex to opposite side midpoint was equal distant.