Posted by Lily on .
Find the distance from the point (5, 6) to the line x + y = 3
I don't get the x+y=3 mean

Math grade 9 
Damon,
Well, it means the shortest distance, that along a perpendicular from the point to the line.

Math grade 9 
Damon,
slope of the line y = x + 3 is 1
so
the slope of the perpendicular is 1/1 = 1 
Math grade 9 
Damon,
then find the equation of that perpendicular through the given point
y = 1 x + b
goes through our point (5,6)
so
6 = 5 + b
b = 1
so the equation of our perpendicular is
y = x + 1 
Math grade 9 
Damon,
Now where does that hit our original line?
y = x + 3
y = x + 1
 add them
2 y = 4
y = 2
then x = 1
so at
(1,2)
NOW I have an easy problem that I bet you know how to do.
Find the distance from
(5,6) to (1,2) 
Math grade 9 
Damon,
d^2 = (15)^2 + (26)^2
d^2 = 16 + 16
d^2 = 2*4^2
d = 4 sqrt 2 
Math grade 9 
Reiny,
If (a,b) is a point not on the line
Ax + By + C = 0
then the shortest distance from the point to the line is
aA + bB + C/√(A^2+b^2)
shortest distance
= 5(1) + 6(1) 3/√(1^2+1^2)
= 8/√2
which rationalizes to 4√2 
Math grade 9 
Damon,
Are they supposed to know that Reiny ?

Math grade 9 
Reiny,
not sure about grade 9, but I just checked an old grade 10 text from Ontario, and I used to teach it in 10
The proof followed your method using a general case. 
Math grade 9 
Damon,
I think you had best learn that recipe Lily.