1
$\begingroup$

Right triangle

C _ _ _ _ _ _ B
  \        |_|
   \         |     A = (ax,ay)
    \        |     B = (bx,by)
     \       |     BC = L
      \      |     C = (?,?)
       \     |
        \    |
         \   |
          \  |
           \ |
            \|
             A

Okay so lets say you are given coordinates of points A, B and the length of BC. How do you find the coordinates of vertex C knowing it is a right triangle?

With the information I know I can get length AB(distance formula), length AC(Pythagorean), and then all the angles using law of sines but still at a blank on how to find C.

Edit: The sides are not necessarily parallel to or on any axis. The triangle may be rotated in any direction on the Cartesian plane.

Any guidance would be really helpful! I feel like this is a simple geometry problem and I have tried searching the internet but only find problems similar to this one and I can't figure it out.

Again any help would be really appreciated :) Thanks in advance!

$\endgroup$
0
$\begingroup$

Let $C(c_x,c_y)$. Then, we have two equations about $c_x,c_y$ :

$$\vec{BA}\cdot\vec{BC}=0\iff (a_x-b_x)(c_x-b_x)+(a_y-b_y)(c_y-b_y)=0$$$$BC=L\iff (c_x-b_x)^2+(c_y-b_y)^2=L^2$$ If you have $c_x\lt b_x$, then you can get $c_x,c_y$ by solving these.

$\endgroup$
  • $\begingroup$ Wow thank you for your answer but what if cx >= bx? $\endgroup$ – Chloe Carmichael Nov 24 '15 at 19:05
  • $\begingroup$ @ChloeCarmichael: Note that we have two possible points for $C$ from your conditions. To determine the coordinate of $C$, we need one more condition. $\endgroup$ – mathlove Nov 24 '15 at 19:15
  • $\begingroup$ Yeah I see your point now. Thank you very much @mathlove ! I think I can make the problem I want to use this on work with those conditions. $\endgroup$ – Chloe Carmichael Nov 24 '15 at 19:21
  • $\begingroup$ @ChloeCarmichael: You are welcome. $\endgroup$ – mathlove Nov 24 '15 at 19:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.