SOLUTION: Find the exact directed distance from the line 3x-2y=5 to the point (-5,6). Thanks!

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Question 231159: Find the exact directed distance from the line 3x-2y=5 to the point (-5,6).
Thanks!
Answer by user_dude2008(1862) About Me  (Show Source):
You can put this solution on YOUR website!
3x-2y=5
-2y=5-3x
-2y=-3x+5
y=3/2x-5/2
y=1.5x-2.5

Solved by pluggable solver: Finding a distance between a point given by coordinates (x, y) and a line given by equation y=ax+b
We want to find the perpendicular distance between a point given by coordinates (-5,6)
and a line given by equation y=1.5%2Ax%2B-2.5

First, let's draw a diagram of general situation with point P (xo, yo) and
line L: y= a.x + b. The required distance is PC. (in the diagram below)





Methodology
We will first find the vertices of the triangle in order to get the side lengths and then by applying
Sine Rule on right angle triangle PAB and PBC we will calculate the desired distance PC.


Step1
Calculation of the vertices of triangle PAB:

Draw a vertical line passing through the point 'P'. This line x=-5 will cut the given line 'L'
at point 'A'. The X coordinate of A(x1) will be same as xo=-5. To find the Y-coordinate of
'A' we will use the fact that point 'A' lies on the given line 'L' and satisfies the equation
of the line 'L' .
Now, plug this x1=-5 in to the equation of line: y=1.5*x+-2.5
y1=1.5%2A-5+%2B-2.5
y1=-10

Hence, Point (A)(x1=-5,y1=-10)


Similarly,
Draw a horizontal line passing through the point 'P'. This line y=6 will cut the given line 'L'
at point 'B'. The Y coordinate of B(y2) will be same as yo=6. To find the X-coordinate of
B we will use the fact that point 'B' lies on the given line 'L' and satisfies the equation
of the line 'L' .
Now, plug this y2=6 in to the equation of line: y=1.5*x+-2.5
6=1.5%2Ax2%2B-2.5
x2=+%286--2.5%29%2F1.5
x2=5.66666666666667

Hence, Point (B)(x2=5.66666666666667,y2=6)


Now, we have all the vertices of the triangle PAB


Step2
Calculation of the side lengths using distance formula:

d=sqrt%28%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2%29


Hence, The side lengths PA, PB and AB are
PA=16
PB=10.6666666666667
AB=19.2296068024746


Step3
Apply Sine rule on common angle B in triangle PAB and triangle PBC.
Both triangle PAB and triangle PBC are right angle triangle and points 'A', 'B' and 'C' lay on the given line L.

Sin%28B%29=+AP%2FAB=PC%2FBP

PC=%28AP%2ABP%29%2FAB=+8.87520313960367


PC is the required perpendicular distance of the point P (-5, 6) from line given
lineL1: y=1.5*x+-2.5.


For better understanding of this concept, look at the Lesson based on the above concept.
Lesson