How do you find the coordinates of a point given its distance from the origin?

To find the coordinates of a point given its distance from the origin, use the Pythagorean theorem or trigonometry.

When you know the distance of a point from the origin, you can think of this distance as the hypotenuse of a right-angled triangle. The origin is at (0,0), and the point lies somewhere on a circle centred at the origin with a radius equal to the given distance. If you know the angle the point makes with the positive x-axis, you can use trigonometry to find the exact coordinates.

For example, if the distance (radius) is \( r \) and the angle \( \theta \) is known, the coordinates \((x, y)\) can be found using:
\[ x = r \cos(\theta) \]
\[ y = r \sin(\theta) \]

If you don't know the angle, but you know the x or y coordinate, you can use the Pythagorean theorem. For instance, if you know the x-coordinate \( x \) and the distance \( r \), you can find the y-coordinate using:
\[ y = \sqrt{r^2 - x^2} \]
Similarly, if you know the y-coordinate \( y \) and the distance \( r \), you can find the x-coordinate using:
\[ x = \sqrt{r^2 - y^2} \]

Remember, there can be two possible points (one in each quadrant) that satisfy these conditions, so consider both positive and negative values for the square root. This method helps you locate the point accurately on the coordinate plane.