Question

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?

Answer 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.