find the distance from the point (9-2) to the line y =3/2x+4

Question

find the distance from the point (9-2) to the line y =3/2x+4

Answer 1

distance = square root of 117

= about 10.81665383 or about 10.8

find the equation of the perpendicular line through (9,-2)

It has slope =-2/3

y=(-2/3)x + b, plug in y=-2 and x=9 to solve for b

-2 =(-2/3)(9) + b

b = 4 or the point (0,4)

calculate the distance from (0,4) to (9,-2)

d^2 = 81 + 36 = 117

d = sqr117 = about 10.8

Answer 2

Shortest line from (9,2) is line perpendicular to 3/2x + 4.

Find equation of that line.

slope would be -2/3 as it is perpendicular,

so y + 2 = -2/3(x - 9)

3y + 6 = -2x + 18

3y = -2x + 12

y = -2/3 x + 4 is equation of the perpendicular line

We can use these two equations to find where the perpendicular and original line intersect.

We can equate two expressions in y and solve for x

3/2x + 4 = -2/3x + 4

9x + 24 = -4x + 24

x = 0

substituting into either equation y = 4

so we have two points on the perpendicular line (9,-2) and (0,4)

the distance between those two points is the distance from the point to the line

sqrt( (0-9)^2 + (4--2)^2) = sqrt(117) = 10.8.

Answer 3

d=sqrt((x2-x1)²+(y2-y1)²)