SOLUTION: Find the distance from point A(15,−21) to the line 5x+2y = 4. Round your answer to the nearest tenth.

Algebra ->  Angles -> SOLUTION: Find the distance from point A(15,−21) to the line 5x+2y = 4. Round your answer to the nearest tenth.      Log On


   

Question 1146601: Find the distance from point A(15,−21) to the line 5x+2y = 4. Round your answer to the nearest tenth.
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Find the distance from point A to the given line
A(15,-21),
5x%2B2y=4.....first, write in slope intercept form

2y=-5x%2B4
y=-%285%2F2%29x%2B2
The slope of your GIVEN line is -5%2F2.
The slope of the line you want to find (perpendicular line) must have slope inverse reciprocal which is 2%2F5.
Line you want:
y-%28-21%29=%282%2F5%29%28x-15%29, using point-slope form
y%2B21=%282%2F5%29x-6
highlight_green%28y=%282%2F5%29x-27%29

WHY? The two lines intersect at some point which you will find.
Then use Distance formula to find the length from the intersection point to point A(15,-21).
You are first looking for the line perpendicular to the given one which contains point A.

Now find intersection point of y=-%285%2F2%29x%2B2 and y=%282%2F5%29x-27. So, all you need is equal right sides.
-%285%2F2%29x%2B2+=+%282%2F5%29x-27
27%2B2+=+%282%2F5%29x%2B%285%2F2%29x
29+=+%2829+x%29%2F10
290+=+29+x
x=290%2F29
x=10
go to y=-%285%2F2%29x%2B2, plug in x=10 and solve for y
y=-%285%2F2%2910%2B2
y=-5%2A5%2B2
y=-25%2B2
y=-23

intersection point is at (10,-23)

now use Distance formula to find the length from the intersection point to point A
A(15,-21) = (x%5B1%5D,y%5B1%5D)
(10,-23) = (x%5B2%5D,y%5B2%5D)

d=sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29
d=sqrt%28%2810-15%29%5E2%2B%28-23-%28-21%29%29%5E2%29
d=sqrt%28%28-5%29%5E2%2B%28-23%2B21%29%5E2%29
d=sqrt%2825%2B%28-2%29%5E2%29
d=sqrt%2825%2B4%29
d=sqrt%2829%29=> exact solution
d=5.4=>to the nearest tenth

Answer by greenestamps(13216) About Me  (Show Source):
You can put this solution on YOUR website!


The response from the other tutor shows a detailed solution to the problem.

You should be able to understand all the steps required for the solution; and you should be able to solve a similar problem by that method.

However, in practice, there is a concise formula for finding the (shortest) distance from a given point to a given line. When the equation of the line is in the form Ax+By+C=0, then the distance from a point (p,q) to the line is

abs%28%28Ap%2BBq%2BC%29%2Fsqrt%28A%5E2%2BB%5E2%29%29

In the required form, the equation in this problem is 5x+2y-4 = 0. Then the distance from (15,-21) to that line is