If I have a right $\triangle ABC$ with $B$ being the right angle and length $AB = 50$ and length $BC = 50$. Based on the Cartesian coordinate system if I wanted to move up the Y axis the length of the hypotenuse of this right triangle my book says I should calculate $\frac{\text{length}}{\sqrt{2}}$.

I don't understand the concept. I know the length of the hypotenuse of a right triangle is $\sqrt{2}$, but I don't understand the rationale behind dividing the length by $\sqrt{2}$ to find how many units up on the y axis to move.

EDIT. I forgot to include the position of the triangle. If the right $\angle B$ is at the coordinate $(0, 0)$. So side $AB$ is on the Y axis and side $BC$ is on the X axis.

EDIT. I apologize. I'm having trouble in understanding my homework. The question is how to calculate how many units up the Y axis to move if the hypotenuse length of a right triangle (ex. $50$)? My book says divide the hypotenuse by $\sqrt{2}$ but I don't understand why. Please disregard the length values in the first paragraph. All that is given is the hypotenuse length.

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    $\begingroup$ If you go $Q$ units up the axis, then the length of the hypotenuse is $Q$ times $\sqrt2$. So the length of the hypotenuse is $\sqrt2$ times how far you go up the axis. So how far you go up the axis is the length of the hypotenuse divided by $\sqrt2$. $\endgroup$ – Gerry Myerson May 17 '14 at 13:15
  • $\begingroup$ Is the question actually how far up the $y$ axis to move in order to travel from point $B$ (at $(0,0)$) to point $A$? That is, you are trying to find the $y$-coordinate of $A$? Then the answer is the length of side $AB$, and the previous comment shows how you compute it. $\endgroup$ – David K May 17 '14 at 13:36
  • $\begingroup$ @davidK I'm trying to figure out how far up to move on the y axis from the length of AC the hypotenuse. So if the hypotenuse is 50 how far up do i move on the y axis. $\endgroup$ – Jessica M. May 17 '14 at 20:56
  • $\begingroup$ @GerryMyerson I don't fully understand how you initially come up with the formula that the length of the hypotenuse is Q times square root of 2. Is that information from sine, cosine, and tangent or Pythagorean theorem or some other general rule? $\endgroup$ – Jessica M. May 17 '14 at 22:56
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    $\begingroup$ Thanks. Let me suggest that, now that you understand what's going on, you write it up and post it as an answer, yourself. It's good practice, and then when the software allows it you can accept your own answer. May seem like a strange thing to do, but the website actually encourages people to answer their own questions after they have figured them out. $\endgroup$ – Gerry Myerson May 19 '14 at 0:56

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