SOLUTION: How far is the point (5,3) from the line y = 4x+1? Hint: the point must lie on a line perpendicular to the given line. Use the distance formula for two points in a plane.

Algebra ->  Graphs -> SOLUTION: How far is the point (5,3) from the line y = 4x+1? Hint: the point must lie on a line perpendicular to the given line. Use the distance formula for two points in a plane.      Log On


   

Question 112353This question is from textbook
: How far is the point (5,3) from the line y = 4x+1?
Hint: the point must lie on a line perpendicular to the given line.
Use the distance formula for two points in a plane. This question is from textbook

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
First, find the line perpendicular to your original line that goes through the point (5,3).
Perpendicular lines have slopes that follow this relationship,
m%5B1%5Dm%5B2%5D=-1
Their slopes are negative reciprocals.
Your original line slope is 4.
Your perpendicular line slope is then (-1/4).
Using the point-slope form of the line, your perpendicular line is,
y-y%5B1%5D=m%28x-x%5B1%5D%29
y-3=%28-1%2F4%29%28x-5%29
y=%28-1%2F4%29x%2B5%2F4%2B3
y=%28-1%2F4%29x%2B5%2F4%2B12%2F4
y=%28-1%2F4%29x%2B17%2F4
+graph%28+300%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C+4x%2B1%2C+%28-1%2F4%29x%2B17%2F4%29+
Now that you have the equation for the two lines, you can find the intersection point where the two lines meet.
From there you can calculate the distance from the intersection point to (5,3) to get the final answer.
1.y+=+4x%2B1
2.y=%28-1%2F4%29x%2B17%2F4
Substitute equation 1 into equation 2 and solve for x.
2.y=%28-1%2F4%29x%2B17%2F4
4x%2B1=%28-1%2F4%29x%2B17%2F4
%2816%2F4%29x%2B%281%2F4%29x=17%2F4-1
%2817%2F4%29x=17%2F4-4%2F4
%2817%2F4%29x=13%2F4
x=13%2F17
From 1,
1.y+=+4x%2B1
y+=+4%2813%2F17%29%2B1
y+=+52%2F17%2B17%2F17
y+=+69%2F17
Now you have found the intersection point.
Use the distance formula to find the distance between the intersection point and the point (5,3)
D%5E2=%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2
D%5E2=%2813%2F17-5%29%5E2%2B%2869%2F17-3%29%5E2
D%5E2=%2813%2F17-85%2F17%29%5E2%2B%2869%2F17-51%2F17%29%5E2
D%5E2=%28-72%2F17%29%5E2%2B%2818%2F17%29%5E2
D%5E2=%285184%2F289%29%2B%28324%2F289%29
D%5E2=%285508%2F289%29
D%5E2=%285508%2F289%29
D=4.37