# Find distance between two points on Cartesian plane

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 $$\overline{BC}$$?

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

## 1 Answer

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 $$\overline{BC}$$.