Answer:
The distance is [tex]3\sqrt{2}\ units[/tex]
Step-by-step explanation:
step 1
Find the slope of the give line
we have
y=x+2
so
the slope m is equal to
m=1
step 2
Find the slope of the perpendicular line to the given line
Remember that
If two lines are perpendicular, then their slopes are opposite reciprocal of each other
so
we have
m=1 -----> slope of the given line
therefore
The slope of the perpendicular line is equal to
m=-1
step 3
With m=-1 and the point (8,4) find the equation of the line
y-y1=m(x-x1)
substitute
y-4=-(x-8)
y=-x+8+4
y=-x+12
step 4
Find the intersection point lines y=x+2 and y=-x+12
y=x+2 -----> equation A
y=-x+12 ----> equation B
Adds equation A and equation B
y+y=2+12
2y=14
y=7
Find the value of x
y=x+2 -----> 7=x+2 -----> x=5
The intersection point is (5,7)
step 5
Find the distance between the point (8,4) and (5,7)
we know that
The distance from the point (8,4) to the line y=x+2 is equal to the distance from the point (8,4) to the point (5,7)
Find the distance AB
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
substitute
[tex]d=\sqrt{(7-4)^{2}+(5-8)^{2}}[/tex]
[tex]d=\sqrt{(3)^{2}+(-3)^{2}}[/tex]
[tex]d=\sqrt{18}[/tex]
[tex]d=3\sqrt{2}\ units[/tex]
see the attached figure to better understand the problem
