Step 1: Find the gradient of line AB.

The gradient \(m\) of line AB is given by:

\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 1}{3 - (-1)} = \frac{3}{4}\)

Step 2: Find the gradient of line BC.

Since line BC is perpendicular to AB, its gradient is the negative reciprocal of \(\frac{3}{4}\):

\(m_{BC} = -\frac{4}{3}\)

Step 3: Find the equation of line BC.

Using point B (3, 4) and the gradient \(-\frac{4}{3}\), the equation of line BC is:

\(y - 4 = -\frac{4}{3}(x - 3)\)

Simplifying:

\(y = -\frac{4}{3}x + 8\)

Step 4: Find the x-coordinate of C.

Since C is on the x-axis, \(y = 0\):

\(0 = -\frac{4}{3}x + 8\) \(\frac{4}{3}x = 8\) \(x = 6\)

Step 5: Find the distance AC.

Coordinates of A are (-1, 1) and C are (6, 0). Using the distance formula:

\(AC = \sqrt{(6 - (-1))^2 + (0 - 1)^2} = \sqrt{7^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50} = 7.071\)

Thus, the distance AC is 7.071.