find the distance from the point (8, 4) to the line y =
x+ 2.

The distance is

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

substitute

see the attached figure to better understand the problem

ok

step-by-step explanation: