SOLUTION: Find the distance from (6,-3)to the line defined by y= 2x-5. Express as a radical or a number rounded to the nearest hundreth

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Question 849870: Find the distance from (6,-3)to the line defined by y= 2x-5. Express as a radical or a number rounded to the nearest hundreth
Found 3 solutions by Alan3354, Fombitz, Edwin McCravy:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Step 1, find the line perpendicular to y = 2x-5 thru (6,3), point A.
Step 2, find the intersection of the 2 lines, point B.
Step 3, find the distance AB.
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Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Do you mean the minimum distance to the line?
If so, the minimum distance is a perpendicular line going through the point.
A perpendicular line would have the form,
y=-%281%2F2%29x%2Bb
Use the point to solve for b.
-3=-%281%2F2%29%286%29%2Bb
-3=-3%2Bb
b=0
.
.
.
Find the intersection of the two lines.
y=-%281%2F2%29x
y=2x-5
Set them equal to each other.
-%281%2F2%29x=2x-5
-x=4x-10
-5x=-10
x=2
Then from either equation, solve for y.
y=-%281%2F2%29%282%29=-1
Now find the distance from (6,-3) to (2,-1) using the distance formula,
D%5E2=%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2
D%5E2=%282-6%29%5E2%2B%28-3-%28-1%29%29%5E2
D%5E2=%28-4%29%5E2%2B%28-2%29%5E2
D%5E2=16%2B4
D%5E2=20
highlight%28D=2sqrt%285%29%29

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
They did it the long way.  

The formula for the distance from a point to a line is

d%22%22=%22%22abs%28Ax%5B1%5D%2BBy%5B1%5D%2BC%29%2Fsqrt%28A%5E2%2BB%5E2%29

where the equation of the line is Ax+By+C=0 and the point is (x1,y1)

The point is (x1,y1) = (6,-3)
The line's equation is y=2x-5. we must get that in the form
Ax+By+C=0

      y=2x-5.  We subtract the entire right side from both sides
 
-2x+y+5=0

A=-2, B=1, C=5, (x1,y1) = (6,-3)

d%22%22=%22%22abs%28Ax%5B1%5D%2BBy%5B1%5D%2BC%29%2Fsqrt%28A%5E2%2BB%5E2%29

d%22%22=%22%22abs%28-2%286%29%2B1%28-3%29%2B%285%29%29%2Fsqrt%28%28-2%29%5E2%2B%281%29%5E2%29

d%22%22=%22%22abs%28-12-3%2B5%29%2Fsqrt%284%2B1%29

d%22%22=%22%22abs%28-10%29%2Fsqrt%285%29

d%22%22=%22%2210%2Fsqrt%285%29

We can rationalize the denominator by multiplying by sqrt%285%29%2Fsqrt%285%29.

d%22%22=%22%2210%2Fsqrt%285%29%22%22%2A%22%22sqrt%285%29%2Fsqrt%285%29%22%22=%22%2210sqrt%285%29%2F5%22%22=%22%222sqrt%285%29.

That's about 4.47, so let's draw the graph as a check:



That green line looks like it's about 4.47 units in length.

Edwin