Calculus

A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (8,4).
Write the length L of the hypotenuse as a function of x.

Completely stuck came up with the sqrt(x^2+Y^2)=L, not sure what to do? can anyone guide me in the right direction?

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asked by Taryn
  1. any line through (8,4) in the first quadrant is

    y-4 = m(x-8)

    So, if the line in question has slope m, the x-intercept is 8-(4/m) and the y-intercept is 4-8m

    So,

    L = √x^2+y^2 = √((8-(4/m))^2 + (4-8m)^2)
    = -4/m √((m^2+1)(2m-1)^2)

    Of course, there are other ways to characterize the line, but something needs to be specified besides just a point on it.

    Check: If the slope is, say, -1, the intercepts are both 12, so we know L = 12√2.
    Our formula says L=12√2
    So, it checks out at at least one point.

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    posted by Steve

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