Math grade 9

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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 -

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

  • Math grade 9 -

    slope of the line y = -x + 3 is -1
    so
    the slope of the perpendicular is -1/-1 = 1

  • Math grade 9 -

    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 -

    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 -

    d^2 = (1-5)^2 + (2-6)^2
    d^2 = 16 + 16
    d^2 = 2*4^2
    d = 4 sqrt 2

  • Math grade 9 -

    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 -

    Are they supposed to know that Reiny ?

  • Math grade 9 -

    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 -

    I think you had best learn that recipe Lily.

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