Since AC is perpendicular to AB, the slope of AC is the negative reciprocal of the slope of BC. The slope of BC is -5, so the slope of AC is 1/5. We can use the point-slope form of a line to find the equation of AC:
y - 6 = (1/5)(x - 3)
Simplifying, we get:
y = (1/5)x + 27/5
To find the coordinates of point C, we need to find where BC and AC intersect. We can do this by setting the equations of BC and AC equal to each other and solving for x:
-5x + 47 = (1/5)x + 27/5
Multiplying both sides by 5, we get:
-25x + 235 = x + 27
Simplifying, we get:
26x = 208
x = 8
Now that we know x = 8, we can plug it into either equation to find y:
y = -5(8) + 47 = 7
Therefore, point C has coordinates (8, 7).