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