How to find shortest distance from using equation y=1/2 + 4 and point (7,17)

Expert Answers
embizze eNotes educator| Certified Educator

sciencesolve used the formula: given a line in General form `Ax+By+C=0` and a point (m,n), then the distance from the point to the line is given by: `(|Am+Bn+C|)/sqrt(A^2+B^2)` , which is great if you can remember the formula.

If you cannot recall the formula, you can proceed as follows:

The shortest distance will be the perpendicular segment from the point to the line. We need a line perpendicular to the given line, through the given point.

The line has a slope of `1/2` , so the line perpendicular to it has slope -2.

Now the perpendicular line has slope -2 and goes through the point (7,17), so the equation of the perpendicular line is `y=-2x+31`

The two lines intersect when `1/2x+4=-2x+31==> x=54/5,y=47/5`

Now we need only find the length of the segment from (7,17) to `(54/5,47/5)` using the distance formula:


sciencesolve eNotes educator| Certified Educator

You need to remember that the shortest distance from the given point to the given line `1/2 x - y + 4 = 0`  is the perpendicular dropped from the point to the line.

`d = |(1/2)*7 - 1*17 + 4|/(sqrt((1/2)^2 + (-1)^2))`

`d = |7 - 34 + 8|/(2sqrt(1/4 + 1))`

`d = |15-34|/(2sqrt(5/4))`

`d = |-19|/(2sqrt5/2) =gt d = 19/sqrt5`

`d = (19sqrt5)/5`

Hence, evaluating the shortest distance from the point (7,17) to the line `(1/2)x - y + 4 = 0=> d = (19sqrt5)/5.`