# How to find the third coordinate of a triangle with specific conditions

I am writing a physics paper relating the motion of objects using Loedel diagrams and standard trigonometry.

I am attempting to prove that an object's velocity can be found by using a specific type of triangle, and it has led me to this problem for which I need assistance:

Using a standard Cartesian coordinate system, and for a given triangle $abc$, with angles $A,B,C$, where $a$ is opposite to $A$, $b$ opposite $B$, and $c$ opposite $C$:

If side $c$ of the triangle is bounded at coordinates $(0,0)$ and $(1,0)$, and angles $A$ and $B$ are acute angles, then:

Find the equation for the $x$- and $y$-coordinates of the third point of the triangle such that for any given angle $A$, $a = h/b$, where $h$ is the height of the triangle, and therefore:

$$\sin A = a$$

By the sine law, we have $$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}.$$ Now, since $a = \sin A$, it follows that $c = \sin C$. It is given that $c = 1$, so $C = \frac \pi 2$, i.e., $abc$ s a right-angle triangle.
• The answer is right, and both $A$ and $B$ are acute angles, while $C=90°$. Commented Dec 31, 2017 at 22:45