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Q&A #1895

Teachers' Lounge Discussion: Distance from a point to a line.

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From: Loyd

To: Teacher2Teacher Public Discussion
Date: 2002040422:56:30
Subject: Re: Distance from a point to a line

The problem that is in most Calculus books for finding the distance from a point to a line involves a knowledge of vectors. For those whose knowledge is limited to algebra, we can find the distance from a point to a line by the following steps: 1. The shortest distance will be a perpendicular to the given line. This can be found by knowing that if two lines are perpendicular, then the slopes are the negative reciprocal of each other. 2. Once we have the two lines, we can solve them to find the point of intersection of the two lines. 3. Using the given point and the point of intersection, we can then use the distance formula to find the distance from the given point to the intersection. Suppose we have a point(j,k) and a given line that has the form y=mx+b . Find the shortest distance between the point and the line. This of course, will be a perpendicular line through the point (j,k). The slope of a perpendicular line is -1/m. This means that the equation for the perpendicular line is y=(-1/m)x + d. Substituting the point into the perpendicular line we have: k=(-1/m)j + d. Solving for d, we obtain: k + j/m = d. Therefore, the equation of the perpendicular becomes: y=(-1/m)x + k+j/m and the given line is: y=mx+b To find the point of intersection, solve for x and y. y=mx+b y=(-1/m)x + d Thus, mx+b=(-1/m)x +d which when solved for x yields: x=[m(d-b)]/[m^2+1]. Substitute this x value into y=mx+b and we obtain y=(1/2)[m(d-b)/(m^2+1)]+b Example: Given Y=(1/2)x + 4 and a point (5,6). Find the line perpendicular to the given line and which passes through the point (5,6)=(j,k) y=(-2/1)x + 6+5/(1/2) y=(-2/1)x + 16 Solve the following pair of equations to find the intersection: Y=(1/2)x + 4 y=(-2/1)x + 16 Solving the last two equations with the TI-83 will speed the process, but you can check by setting the last two equations equal and finding x. Then you can solve for y. The resuslt will be: x=4.8 and y= 6.4 or if you prefer, 24/5 and 32/5. The distance between the two points will be the solution to; sqrt[(5-4.8)^2 + (6-16)^2] which is a little more than 12.

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