# What are the coordinates of the points a and b and minimize the length of the hypotenuse of a right triangle that is formed in the first quadrant by the x-axis, the y-axis, and a line through the point (1,2) where point a is at (0,y) and point b is at (x,0)?

##### 1 Answer

Suppose that the base point along the Y-axis is some scaled factor,

That is the point on the Y-axis #(0,y_a)# is at

or

The point on the X-axis (x_b, 0) must satisfy the equivalent slope ratios:

The square of the ladder length based on the scaling factor can be expressed as

which simplifies to

We want to minimize

Since

and we can simplify our effort with the same results by minimizing

Setting the derivative to zero to find the minimum length:

multiplying by

factoring out

Recalling that

we have

and

No sense stopping now.

From our earlier formula

Congratulations to anyone who made it this far!

I hope there is a simpler method, but I couldn't find it.

I also hope that this is somewhere approaching correct and I didn't mess up.