Question 1197253
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Let's find the slope of the line through (5,3) and (2,-3)
(x1,y1) = (5,3)
(x2,y2) = (2,-3)


m = slope


{{{m = (y[2] - y[1])/(x[2] - x[1])}}}


{{{m = (-3 - 3)/(2 - 5)}}}


{{{m = (-6)/(-3)}}}


{{{m = 2}}}


Now apply point-slope form
y-y1 = m(x - x1)
y-3 = 2(x - 5)
y-3 = 2x - 10
y = 2x - 10+3
y = 2x - 7
This is in y = mx+b form


Let's get everything to one side
y = 2x - 7
0 = 2x - 7 - y
0 = 2x - y - 7
2x-y-7 = 0
This is now in the form Ax+By+C=0 with A = 2, B = -1, C = -7


Now we'll use the formula discussed here
<a href = "https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line">https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line</a>
I'll use slightly different notation
The A,B,C will be the same
Instead of (x0,y0) I'll refer to the point not on the line as (p,q)


The distance from (p,q) to the line Ax+By+C = 0 is {{{d = (abs(Ap+Bq+C))/(sqrt(A^2+B^2))}}} where point (p,q) is not on the line mentioned.


We'll plug in A = 2, B = -1, C = -7
And also p = 12 and q = 7 from the point (12,7)
{{{d = (abs(Ap+Bq+C))/(sqrt(A^2+B^2))}}}


{{{d = (abs(2*12+(-1)*7+(-7)))/(sqrt(2^2+(-1)^2))}}}


{{{d = (abs(24-7-7))/(sqrt(5))}}}


{{{d = (abs(10))/(sqrt(5))}}}


{{{d = 10/(sqrt(5))}}}


{{{d = (10*sqrt(5))/(sqrt(5)*sqrt(5))}}} Multiply top and bottom by sqrt(5) to rationalize the denominator


{{{d = (10*sqrt(5))/((sqrt(5))^2)}}}


{{{d = (10*sqrt(5))/5}}}


{{{d = 2*sqrt(5)}}} is the exact distance from (12,7) to the line 2x-y-7 = 0 which goes through (5,3) and (2,-3)


*[tex \Large 2\sqrt{5} \approx 4.472136]


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Verification:
<img src = "https://i.imgur.com/i9IXZkA.png">
I used GeoGebra to make this image. 


The red line 2x-y-7 = 0 was what we found earlier through (5,3) and (2,-3)
The black line x+2y-26 = 0 is perpendicular to the red line, and it goes through point A(12,7)
Use whichever method you prefer to solve that system of 2 equations to find the red and black line intersect at D(8,9)
Then use the distance formula to find the distance from A(12,7) to D(8,9) is exactly {{{2*sqrt(5)}}} units.


Side note: 
{{{2*sqrt(5) = sqrt(2^2)*sqrt(5) = sqrt(4)*sqrt(5) = sqrt(4*5) = sqrt(20)}}}
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