Moving 1 by 1 on a hypotenuse

Question

I have a right triangle. The h represents a projectile that I want to move at a specific speed. No matter the dimensions the right triangle, I want to move the same speed on its h.

I guess I am asking, how do I figure out the change in the x and y axis as I move along the hypotenuse.

Answer 1

First of all, the triangle you have above is not a right triangle. It doesn't pass the Pythagorean Theorem.

But to answer your question anyway, you should use similarity. Call the horizontal leg $B$, the vertical leg $A$, and the hypotenuse $H$ (assuming these do form a right triangle). Recycle these letters for the respective lengths for these line segments as well.

Say you want the hypotenuse to be of length $h = H+k$ instead of length $H$ (where $k$ is some integer), but you don't want to change the shape of the triangle above - that is, you want the newly formed triangle to be similar to the one drawn above. If the horizontal leg of the newly formed triangle is $b$, the vertical leg $a$, then you must have

$\frac{h}{b} = \frac{H}{B} \quad \Longrightarrow \quad b = \frac{hB}{H}$,

$\frac{h}{a} = \frac{H}{A} \quad \Longrightarrow \quad a = \frac{hA}{H}$.

Note that $a,b$ may not be integers, but they do determine a hypotenuse of length $h$, for

$a^2 + b^2 = \left(\frac{hA}{H} \right)^2 + \left(\frac{hB}{H} \right)^2 = \frac{h^2(A^2+B^2)}{H^2} = h^2$.

Answer 2

The solution to this problem that has plagued me so longgg... hours/ days, so simple.

I did a horrible job of explaining the question but here is the solution:

Divide the axis by the hypotenuse to get rate of change.

So in my example:

 Delta X = 10/ 11.6
 Delta Y = 6/ 11.6