The formula for calculating the distance between two points is:
d = sqrt((color(red)(x_2) - color(blue)(x_1))^2 + (color(red)(y_2) - color(blue)(y_1))^2)
Substituting the values from the points in the problem and solveing for a gives:
sqrt(85) = sqrt((color(red)(a) - color(blue)(8))^2 + (color(red)(4) - color(blue)(-5))^2)
sqrt(85) = sqrt((color(red)(a) - color(blue)(8))^2 + (color(red)(4) + color(blue)(5))^2)
sqrt(85) = sqrt((color(red)(a) - color(blue)(8))^2 + 9^2)
sqrt(85) = sqrt((color(red)(a) - color(blue)(8))^2 + 81)
(sqrt(85))^2 = (sqrt((color(red)(a) - color(blue)(8))^2 + 81))^2
85 = (color(red)(a) - color(blue)(8))^2 + 81
85 = a^2 - 16a + 64 + 81
85 - color(red)(85) = a^2 - 16a + 64 + 81 - color(red)(85)
0 = a^2 - 16a + 60
a^2 - 16a + 60 = 0
(a - 10)(a - 6) = 0
Solution 1:
a - 10 = 0
a - 10 + color(red)(10) = 0 + color(red)(10)
a - 0 = 10
a = 10
Solution 2:
a - 6 = 0
a - 6 + color(red)(6) = 0 + color(red)(6)
a - 0 = 6
a = 6
a can be either 6 or 10
