Question

Find the distance from P to l.Line I contains points (0, -3) and (7,4). Point P has coordinates (4,3)

Answer 1

Find the distance from P to l.

Line I contains points (0, -3) and (7,4). Point P has coordinates (4,3)

step 1

Find the slope of line l

we have the points (0, -3) and (7,4)

m=(4+3)/(7-0)

m=7/7

m=1

step 2

Find the equation of the line perpendicular to the line l that passes through the point P

REmember that

If two lines are perpendicular, then the product of their slopes is equal to -1

therefore

the slope of the perpendicular line is

m=-1

Find the equation in point slope form

y-y1=m(x-x1)

we have

m=-1

(x1,y1)=P(4,3)

substitute

y-3=-1(x-4)

y-3=-x+4

y=-x+7

step 3

Find the equation of the line l

we have

m=1 (see the step 1)

and we have the point (0,-3) -----> y-intercept

so

the equation of the line l is

y=x-3

step 4

Find the intersection point line l and its perpendicular line

y=x-3 ------> equation A

y=-x+7 -----> equation B

solve the system of equations

Adds equation A and equation B

2y=-3+7

2y=4

y=2

Find the value of x

substitute in equation A

2=x-3

x=2+3

x=5

therefore

the intersection poin is (5,2)

step 5

Find the distance between point (5,2) and point P(4,3)

Remember that, this distance is the distance from P to l.

the formula to calculate the distance between two points is equal to

$d=\sqrt[]{(y2-y1)^2+(x2-x1)^2}$

we have

(x1,y1)=(5,2)

(x2,y2)=(4,3)

substitute in the formula

$\begin{gathered} d=\sqrt[]{(3-2)^2+(4-5)^2} \\ d=\sqrt[]{1+1} \\ d=\sqrt[]{2\text{ }} \end{gathered}$

therefore

teh answer is

the distance from P to l. is equal to square root of 2