Find the distance of a point 𝑃 𝑥, 𝑦 from the origin.
The distance formula is a useful formula that’s derived from the Pythagorean theorem that helps us find the distance between two points on a pair of axes. The distance formula says that, for two points 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, the distance between these points is the square root of the square of the differences between the 𝑥-coordinates plus the square of the difference between the 𝑦-coordinates. It’s the square root of 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared.
For this question, the first ordered pair is the one at 𝑃. It’s 𝑥, 𝑦. The second ordered pair is the one that corresponds to the origin. That’s zero, zero. So actually, we’re trying to find the distance between the points 𝑥, 𝑦 and zero, zero. Now it doesn’t matter which of these ordered pairs we choose to be 𝑥 one, 𝑦 one and which we choose to be 𝑥 two, 𝑦 two. The formula works either way round as long as we’re consistent.
Let’s see what this looks like. First, let’s choose the ordered pair 𝑥 and 𝑦 to the 𝑥 one, 𝑦 one. Then the origin becomes 𝑥 two, 𝑦 two. Substituting these values into the distance formula, we get that the distance between them is the square root of zero minus 𝑥 all squared plus zero minus 𝑦 all squared. That simplifies to the square root of negative 𝑥 squared plus negative 𝑦 squared.
Remember, when we square a negative number, we always get a positive result. So the distance between the point 𝑃 and the origin is given by the square root of 𝑥 squared plus 𝑦 squared.
Now let’s just check what this would’ve looked like had we chosen our coordinates the other way round. Substituting them into the formula gives us the square root of 𝑥 minus zero all squared plus 𝑦 minus zero all squared. Once again, that simplifies to the square root of 𝑥 squared plus 𝑦 squared. The distance of the point 𝑃 from the origin is the square root of 𝑥 squared plus 𝑦 squared.