Find the distance from the point (5, 6) to the line x + y = 3
I don't get the x+y=3 mean
Well, it means the shortest distance, that along a perpendicular from the point to the line.
slope of the line y = -x + 3 is -1
so
the slope of the perpendicular is -1/-1 = 1
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
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)
d^2 = (1-5)^2 + (2-6)^2
d^2 = 16 + 16
d^2 = 2*4^2
d = 4 sqrt 2
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
Are they supposed to know that 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.
I think you had best learn that recipe Lily.
To find the distance from a point to a line, we can use the formula:
distance = |Ax + By + C| / sqrt(A^2 + B^2)
Here, A, B, and C are the coefficients of the equation of the line, and (x, y) are the coordinates of the given point.
In the equation x + y = 3, the coefficients A and B are both 1, and C is -3 (since we want the equation in the form Ax + By + C = 0).
So, the equation becomes:
distance = |(1)(5) + (1)(6) + (-3)| / sqrt((1)^2 + (1)^2)
Simplifying further:
distance = |5 + 6 - 3| / sqrt(1 + 1)
distance = |8| / sqrt(2)
Now, we just evaluate the expression:
distance = 8 / sqrt(2)
To simplify the expression further, we can multiply both the numerator and denominator by sqrt(2):
distance = (8 / sqrt(2)) * (sqrt(2) / sqrt(2))
distance = 8sqrt(2) / 2
Simplifying the fraction:
distance = 4sqrt(2)
Therefore, the distance from the point (5, 6) to the line x + y = 3 is 4sqrt(2) units.