Find the length of a segment from point (12,-10) to the line y=4x+27

if you could show how to do it that would

The distance from the given point to the given line is 5√17.

Step-by-step explanation:

Recall that the shortest distance from a point to a line is on another line, which in turn is perpendicular to the given line.  Here the given line is y = 4x + 27, and the slope is 4; any line perpendicular to this line has a slope which is the negative reciprocal of 4, which is -1/4.

Find the equation of the line with slope -1/4 passing through the given point (12, -10):

Use the slope-intercept method:  y = mx + b becomes -10 = (-1/4)(12) + b.  Find the y-intercept, b:  -10 = -3 + b.  then b = -7.

The perpendicular line is y = (-1/4)x - 7.

Now we want to find the distance between this given point (12, -10) to the intersection of the perpendicular line y = (-1/4)x - 7 with the given line y = 4x + 27.  In other words we must solve the system of linear equations:

y = (-1/4)x - 7

y = 4x + 27

and to do this we equate these two equations, eliminating y and enabling us to find the x-coordinate of the point of intersection of the two lines.

y = 4x + 27 = y = (-1/4)x - 7

This becomes (4  1/4)x = -34, or

17x      -34

=

 4         1

Cross multiplying, we get 17x = - 136, or x = -8.

Since y = 4x + 27, if x = -8 we get y = 4(-8) + 27, or -32 + 27, or -5.

The point of intersection of the two lines is then (-8, -5).

All we need to do now is to find the distance between (-8, -5) and the given point (12, -10).  The change in x as we go from (-8, -5) to (12, -10) is 20 and the change in y is -5.  The distance we want is found using the Pythagorean Theorem:  (20)^2 + (-5)^2 = d^2, where d represents that distance;

we get d^2 = 400 + 25, or d^2 = 425, or d = +√425, or

d = +√(25)(17), or d +5√17.

The distance from the given point to the given line is 5√17.

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step-by-step explanation: