Distance from a Point to a Line
Date: 08/20/99 at 11:35:31 From: Carolyn Gallagher Subject: Deriving the formula for distance from a point to a line I am a math teacher in Ontario, Canada looking for a straightforward way to derive the formula for the distance from a point to a line abs(Ax + By + C)/sqrt(A^2+B^2) = D Sorry about the form, but you probably get the general idea. This topic is in the new curriculum for tenth grade mathematics as part of the analytic geometry strand. I know that it is easy enough to show specific cases given the point and the equation of the line, but how do I explain the equation to such a young audience? Carolyn
Date: 08/20/99 at 14:15:47 From: Doctor Anthony Subject: Re: Deriving the formula for distance from a point to a line If the equation of the straight line is ax + by + c = 0, and (h,k) is the point from which we require the perpendicular distance, we can prove the result as follows. Assume the line has positive slope and lies in the first quadrant in a convenient position. Let (h,k) be a point in the first quadrant below the given line but with k greater than the y coordinate where the given line cuts the y axis. This diagram will avoid unnecessary complications in explaining the geometry. Draw the perpendicular from P(h,k) to the line. Let the foot of this perpendicular be point N. Draw a horizontal line from P to meet the given line at M. If theta is the angle of slope of the given line then the perpendicular PN is given by PM*sin(theta). The x coordinate of M is found by putting y = k in the equation ax + by + c = 0 ax + bk + c = 0 so ax = -(bk+c) x = -(bk+c)/a Therefore PM = h - [-(bk+c)/a] = [ah + bk + c]/a Now tan(theta) = -(a/b) from equation of line. Then from Pythagoras /| / | / | a hypotenuse = sqrt(a^2+b^2) / | / | sin(theta) = a/sqrt(a^2+b^2) /_____| b and so PM.sin(theta) = [ah+bk+c]/a * a/sqrt(a^2+b^2) PN = [ak+bk+c]/sqrt(a^2+b^2) So the perpendicular distance from (h,k) to the line ax + by + c = 0 is given by ah + by + c ------------- sqrt(a^2+b^2) If this result is positive the point (h,k) is on the same side of the given line as the origin. If the answer is negative then (h,k) is on the opposite side of the line from the origin. This can be used as a check to see on which side of a line a given point lies. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/
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