## LucyLu15 one year ago Does any one know how to find the distance between the point (2,3) and the line 4x + 3y = 10?

1. 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).

2. LucyLu15

Could x=2 @mathstudent55

3. LucyLu15

could that work?

4. Matt.Mawson

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

5. LucyLu15

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

6. mathstudent55

What did you get for the equation of the perpendicular?

7. LucyLu15

i plugged in the one that you had calculated

8. mathstudent55

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

9. mathstudent55

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

10. mathstudent55

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

11. 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.

12. LucyLu15

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

13. mathstudent55

Yes, I have both of those.

14. LucyLu15

for my point of intercept i have .88, 2.16

15. mathstudent55

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

16. LucyLu15

so how will i find the distance?

17. mathstudent55

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

18. mathstudent55

For that distance I get 1.4

19. LucyLu15

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

20. mathstudent55

wlcm