Question 1204346: determine the shortest distance from the point e(1,-4) to the line through points of(-5'2) and g(3,4)
Found 3 solutions by josgarithmetic, mananth, greenestamps:
Answer by josgarithmetic(39616)   (Show Source):
You can put this solution on YOUR website! Find the slope of FG. Find the negative reciprocal of that. Now you have the slope of the line connecting line FG and point E. Find equation of this line connecting E to FG. Remember the line you want contains point E. Do you see what to do to fisnish?
Answer by mananth(16946)  (Show Source):
You can put this solution on YOUR website! Equation of line the line through points of(-5,2) and g(3,4)
use formula
(y-y1)/(y1-y2) = (x-x1)/(x1-x2)
we get
2x - 8y + 26 = 0.....................................(1)
Slope of of line the line through points of(-5,2) and g(3,4)
(y1-y2)/(x1-x2) we get m=1/4
A perpendicular line to fg will have slope -4
Equation of line the line through point(1,-4) m=-4
y=-4x
substitute y in (1)
solve
we get
(-13/17, 52/17)
Distance between (-13/17, 52/17) and (1,-4)
7.06 units The perpendicular distance from E to the line
 .
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
The answer from tutor @Mananth is incorrect. There is an error somewhere in her work.
The method she shows is valid:
(1) Find the equation of the line through the two given points;
(2) Find the equation of the line perpendicular to that given line passing through (1,-4);
(3) Find the point of intersection of the two lines; and
(4) find the distance between (1,-4) and that point of intersection
Do all those steps correctly and you will get the right answer.
But there is a much faster way to solve the problem.
There is an easy formula for finding the shortest distance from a given point to a given line.
If the equation of the given line is Ax+By+C=0 (the equation must be in that form), and the given point is (p,q), then the shortest distance from the point to the line is
For the given problem, the equation of the line through (-5,2) and (3,4), in the required form, is
Then, with the given point (1,-4), the shortest distance from the point to the line is
That gives the shortest distance as 7.276, to 3 decimal places.
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