Distances from points to lines

A closely related problem to the above is that of finding the distance from a point to a line. Our approach is very straightforward: to find the distance from the point P to the line L, first find the line L^⊥ passing through P and perpendicular to L (we can use the point-slope form of a line, y - y_1 = m(x - x_1) for this), then find the point Q where L and L^⊥ intersect. The distance will then be the distance between P and Q, found using the distance formula.

Find the distance from the point (-1, 3) to the line y = 3/4x - 2.

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We first find the line passing through (-1, 3) and perpendicular to y = 3/4x - 2. Using the point-slope formula gives us y - 3 = -4/3 (x + 1).

  

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The intersection of the two lines is easily found to be the point  (44/25, -17/25). Finally, the distance between the points is found with the distance formula:

((-1 - 44/25)^2 + (3 - (-17/25))^2)^(1/2) = ((69/25)^2 + (92/25)^2)^(1/2)
= ((23/25)^2 (3^2 + 4^2))^(1/2)
= 23/2525^(1/2) .
= 23/5

Created by Mathematica  (December 3, 2004)