First, let's consider the equation of that line.

slope = m = (2 - 6)/[(-5) - (-2)] = (-4)/(-3) = 4/3

Next, let's find the y intercept of the line:

y = mx + b

b = y - mx = y - (4x/3)

Since the point (-2,6) is on the line

b = 6 - (4/3)(-2)

b = (26/3)

so that

y = (4x/3) + (26/3)

When x = 3, y = 4 + (26/3) = (38/3)

If we measure the distance from P (3,-4) along the line x = 3, we get

(38/3) + 4 = (50/3)

Similarly, when y = -4

4x/3 = (-4) - (26/3)

4x = (-12) - (26) = -38

x = (-19/2)

so the distance from P to the line measured along the line y = -4 equals 12.5

The three points (3,-4), (-5, -4) and (-5, 2) define a right triangle the length of whose hypotenuse is the length for which we have been searching.

Length of side 1 = Δy = 2 - (-4) = 6

Length of side 2 = Δx = 3 - (-5) = 8

Finally, the length of the hypotenuse is given by the Pythagorean Theorem:

**L = [(6)**^{2} + (8)^{2}]^{½} = 10

If we wish to find the distance along a line perpendicular to line 1 that passes through p

y = (-3x/4) + b

(-4) = (-9/4) + b

b = (-4) + (9/4) = -7/4

Line 2: y = [(-3x)/4] - (7/4)

The two lines intersect at (-5,2) (prove to yourself that this is so).