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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 $\overline{BC}$?

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  • $\begingroup$ what have you tried? $\endgroup$ – MoonKnight Jan 28 '20 at 20:05
  • $\begingroup$ Try looking up the Pythagorean theorem $\endgroup$ – gt6989b Jan 28 '20 at 20:12
  • $\begingroup$ Use Pythagorean theorem. $\endgroup$ – Vasya Jan 28 '20 at 20:12
  • $\begingroup$ are we using a euclidean metric ? $\endgroup$ – user645636 Jan 28 '20 at 20:18
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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 $\overline{BC}$.

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