I have a triangle on the Cartesian plane where I know the following:
$$A = (X_1, Y_1)$$
$$B = (X_2, Y_2)$$
$$C = (X_3, Y_3)$$
$$\angle ABC = 90$$
$$\overline{AB} = x$$
$$\overline{AC} = 2x$$
I know A and B but I don't know C's location.
Can I use these parameters to find distance $\overline{BC}$?
Yes and No.
Since we know the angle $\measuredangle ABC = 90º$, by the Pythagorean Theorem we know that $$(\overline{AB})^2 + (\overline{BC})^2 = (\overline{AC})^2$$
$$(\overline{BC})^2 = (\overline{AC})^2 - (\overline{AB})^2$$
$$(\overline{BC})^2 = (2x)^2 - x^2$$
$$(\overline{BC})^2 = 3x^2 \Leftrightarrow \overline{BC} = \sqrt3|x|$$
Without knowing what $x$ means, we can't find the exact value for $ \overline{BC}$, but we can find a general solution for $\overline{BC}$, where givin any $x$, we know $\overline{BC}$.
