Question

Find the distance from point A(−9,−3) A(−9,−3) to the line y = x−6 . Round your answer to the nearest tenth.

Answer 1

Answer:

8.5 units

Step-by-step explanation:

step 1

Find the slope of the perpendicular line to the given line

$y=x-6$ ---> given line

The slope is $m=1$

Remember that

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

$m_1*m_2=-1$

we have

$m_1=1$

substitute

$(1)*m_2=-1$

so

$m_2=-1$

step 2

Find the equation of the perpendicular line to the given line

The equation in point slope form is

$y-y1=m(x-x1)$

we have

$m=-1$

$(x_1,y_1)=(-9,-3)$

substitute

$y+3=-1(x+9)$  --> equation in point slope form

$y+3=-x-9$

$y=-x-9-3$

$y=-x-12$ ---> equation in slope intercept form

step 3

Find the intersection point between the given line and the perpendicular line to the given line

we have the system of equations

$y=x-6$

$y=-x-12$

Solve the system by graphing

The intersection point is (-3,-9)

see the attached figure

step 4

we know that

The distance between the point A and the point (-3,-9) is the same that the distance between point A and the line y=x-6

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

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

substitute the values

$d=\sqrt{(-9+3)^{2}+(-3+9)^{2}}$

$d=\sqrt{(-6)^{2}+(6)^{2}}$

$d=\sqrt{72}\ units$

simplify

$d=6\sqrt{2}\ units$

$d=8.5\ units$