Do you know how to find the equation of the line through (-10, 9) and (7, -6) (may be called the "two point formula" in your text)? Obviously C must satisfy that equation so you can substitue x and y= that formula into the formula for the distance from (7, -6) to (x, y) and from (-10, 9) to (x, y). Put those into the equation "|BC|= 3/5|AC|" and you will have an equation in x to solve.
Do you know how to find the midpoint of the segment between (-10, 9) and (7, -6)? Do you know the relationship between the slopes of two perpendicular lines? Do you know how to find the equation of a line through a given point with given slope (may be called the "point slope formula" in your text)?