# Shortest distance between Y=-1/2x-3 and the point R(4,5)   Calculate the shortest distance between each point and the given line? Please help step by step with graph.

lfryerda | High School Teacher | (Level 2) Educator

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To calculate the distance from the point R(4,5) to the line `y=-1/2x-3` , we need to find the perpendicular line that intersects R and the line `y=-1/2x-3` .  This intersection happens at a point, let's call it Q.  The slope of `y=-1/2x-3` is `-1/2` , so the slope of RQ is 2.  We can use this with R to find the equation of RQ.

`y=mx+b`

`5=2(4)+b`

`5=8+b`

`5-8=b`

`b=-3`

So the equation of the line RQ is `y=2x-3` .  Noticing that both the original line `y=-1/2x-3` and `y=2x-3` have the same y-intercept (0,-3).  This means that Q is (0,-3), which is the intersection point of the two lines.

The shortest distance from R to the line is the distance from R to Q, which we find using the distance formula:

`d=sqrt{(4-0)^2+(5-(-3))^2}`

`=sqrt{4^2+8^2}`

`=sqrt80`

`=4sqrt5`

The distance from R to the line is `4sqrt5` .  The graph is: