SOLUTION: Find the exact distance from the point D(4,-2) to the line segment joining the points E(1,3) and F(-4,-2)

Question 1137657: Find the exact distance from the point D(4,-2) to the line segment joining the points E(1,3) and F(-4,-2)
Answer by greenestamps(13202) (Show Source): You can put this solution on YOUR website!

The distance from D to the line containing E and F is the length of the line segment from D that is perpendicular to the line.

This particular example is easily solved using equations of perpendicular lines. I'll outline the process; you can fill in the details if you need.

(1) From the coordinates of E and F, we can determine that the equation of the line E and F is y = x+2.
(2) The slope of the line containing E and F is 1; the slope of a line perpendicular to that line is -1.
(3) The line with slope -1 passing through D(4,-2) is y = -x+2.
(4) The intersection of y=x+2 and y=-x+2 is (0,2).
(5) The distance from (4,-2) to (0,2) is 4*sqrt(2).

ANSWER: The distance from D to the line containing E and F is 4*sqrt(2).

In general, there is a concise formula for finding the distance from a given point to a given line.

If the equation of the line is in the form Ax+By+C=0, and the coordinates of the point are (a,b), then the distance from the point to the line is

In this example, after finding the equation y=x+2 for the line containing E and F, put it in the required form:

and plug the numbers into the formula (A=1, B=-1, C=2; (a,b) = (4,-2):

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