Find the distance between the point negative two, four and the point of origin.
To help us understand what this question means, I’ve drawn a little sketch. And actually, what I’ve done is I’ve shown it on a pair of axes. So we actually have two points. We have negative two, four. And we have zero, zero. And that’s zero, zero because that’s the origin. And what we’re trying to do is actually find the distance between those two points. And I’ve represented that with the pink line.
In order to do this, we need to use the distance formula or the distance between two points formula. And the formula is that the distance is equal to the square root 𝑥 two minus 𝑥 one all squared plus 𝑦 two minus 𝑦 one all squared.
What that actually means is the square root of the difference between our 𝑥-coordinates squared plus the difference between our 𝑦-coordinates squared. But why does this give us the distance? Well, the reason we get this formula is because it comes from the Pythagorean theorem.
Well, I’ve drawn a little triangle to help us understand what that would be. So I’ve got a right-angled triangle here. And our distance is actually our hypotenuse. Our difference in 𝑦 is one of our shorter sides. And our difference in 𝑥-coordinates is actually our other shorter side. So therefore, we’ll be able to adapt our Pythagorean theorem to actually give us the hypotenuse, or, in this instance, the distance.
Okay, great! Let’s use the formula now to calculate the distance between our point and the point of origin. So first of all, I’ve labeled our coordinates. And I’ve done this so we can know what we’re gonna put back into our formula without making mistakes. And next, we’re actually gonna substitute these values into our formula.
So we have that the distance is equal to the square root of zero minus negative two all squared. And that’s because that is 𝑥 two, which is zero, minus 𝑥 one, which is negative two. And then that’s all squared. And this is gonna be plus zero minus four all squared. And this is because this is our 𝑦 two and 𝑦 one values. So this is gonna give us that the distance is equal to the square root of two squared plus negative four squared.
And as you can see at this point, we’re actually squaring the differences of each of our 𝑥-coordinates and 𝑦-coordinates. So this would actually mean that it wouldn’t matter which way round we have the points. So if we had our 𝑥 one and 𝑦 one switched with our 𝑥 two and 𝑦 two, it would still work because we’re actually squaring the difference. So the answer will always be positive.
So we can now carry on and say that the distance is equal to root 20. Okay, so is this the final answer? Oh yes, we have got an answer here. But what I always say is if you get a surd answer, make sure you simplify where you can. So we’re gonna try and simplify this further.
In order to simplify this further, we’re gonna use this surd rule, which says that root 𝑎 multiplied by root 𝑏 is equal to root 𝑎𝑏, remembering that we actually want 𝑎 or 𝑏 to be the highest square number factor of 20, which means that we can say that root 20 is equal to root four multiplied by root five. And we can’t simplify that further.
So therefore, we can say that the distance between the point negative two, four and the point of origin will be equal to two root five. And in this question, we just don’t have any units. So what we can often say instead would be that it’s two root five length units.