Does any one know how to find the distance between the point (2,3) and the line 4x + 3y = 10?

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- anonymous

Does any one know how to find the distance between the point (2,3) and the line 4x + 3y = 10?

- katieb

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- mathstudent55

First, find the equation of the line perpendicular to the given line that passes thorugh (2, 3).
Once you have the equation of the perpendicular, solve a system of equations of the two equations to find where the lines intersect.
Then, find the distance between the point of intersection and point (2, 3).

- anonymous

Could x=2 @mathstudent55

- anonymous

could that work?

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- anonymous

Ok this is a big pain to solve, but here it is.
First put the equation of the line into y = m*x+b format
y = (4/3)x + 3.333
The perpendicular line will have slope -3/4 and passes through (2,3) so it's equation will be y = (-3/4)x+4.5
The two lines intersect at (.56,4.08)
The triangle with tip (.56,4.08) and right point at (2,3) has base 1.44 and height of 1.08
\[\sqrt{1.44^{2}+1.08^{2}} = 1.8 \]
1.8 is the answer

- anonymous

when they intersect i get (-2,6)?

- mathstudent55

What did you get for the equation of the perpendicular?

- anonymous

i plugged in the one that you had calculated

- mathstudent55

@Matt.Mawson calculated it, but I think he made a mistake.

- mathstudent55

Let's start from the beginning.
The given line is:
4x + 3y = 10

- mathstudent55

Solve it for y to get the slope:
3y = -4x + 10
y = (-4/3)x + 10/3

- mathstudent55

He made a mistake right in the beginning when he subtracted the 4x from both sides. It should be -4x on the right side, but he wrote 4x.

- anonymous

Yes i get that for my equation as well. and for my perpendicular i calculated y=(3/4)x +1.5

- mathstudent55

Yes, I have both of those.

- anonymous

for my point of intercept i have .88, 2.16

- mathstudent55

Yes, I got the same x and y as you.

- anonymous

so how will i find the distance?

- mathstudent55

The final step is the distance between (0.88, 2.16) and (2,3)

- mathstudent55

For that distance I get 1.4

- anonymous

yes i have that too... Okay. Thank You!

- mathstudent55

wlcm

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