WEBVTT
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So in this lesson, we’re looking at Heron’s formula.
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And what we’re gonna do is learn how to use Heron’s formula to find the area of a triangle.
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And what Heron’s formula is, is a way to find the area of a triangle when the length of all three sides are known.
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And as well as trying to find the area of a triangle using Heron’s formula, we’re also gonna find the area of composite shapes using Heron’s formula.
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Now, you might think it sounds a bit funny, “Heron’s formula.”
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It isn’t, in fact, made up by a heron.
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Heron’s formula, which is also sometimes known as Hero’s formula, is named after the hero of Alexandria who actually devised this formula.
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Okay, before we actually get started and ask some questions to show us how to use it, what we need to do is first of all see what it means.
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So, what Heron’s formula is, is 𝐴, the area, is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑎, 𝑏, and 𝑐 are the sides of our triangle.
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Well, that might be all well and good.
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But what’s 𝑠?
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This isn’t something we’ve seen before.
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Well, 𝑠 is, in fact, the semiperimeter.
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So, what this means is half of the perimeter of our triangle, and we have a formula for this as well.
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And that formula is that 𝑠, the semiperimeter, is equal to 𝑎 plus 𝑏 plus 𝑐 divided by two.
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Well, this makes sense because 𝑎 plus 𝑏 plus 𝑐 will give us our perimeter of our triangle.
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And if we divide it by two, that would be half of the perimeter or the semiperimeter.
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But why would we use Heron’s formula?
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Well, Heron’s formula is really useful because it means we can find the area of a triangle when all we know are all three side lengths.
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So, we don’t have to calculate any other sides or we’d have to find a perpendicular height or we have to find another side using Pythagorean theorem or an angle, nothing like that.
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We can just have the length of all three sides and use it to find the area of our triangle.
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Okay, great.
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So now, we know what Heron’s formula is.
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And we know how to use it.
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Let’s get on and have a look at some examples.
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In our first example, we’re just gonna look at an example where we have to just plug three values into Heron’s formula.
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The area of the triangle whose side lengths are three centimeters, six centimeters, and seven centimeters equals something cm squared.
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So, in this question, what we’re asked to do is find the area of the triangle.
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And what we’re given are the three side lengths.
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And what we know is that if we have the three side lengths and we want to find the area, what we can use is Heron’s formula.
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And what Heron’s formula tells us is that if we have triangle 𝑎𝑏𝑐, then the area is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑠 represents the semiperimeter, which is half the perimeter of our triangle.
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And we’ve got a formula for that; 𝑠 is equal to 𝑎 plus 𝑏 plus 𝑐 over two.
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Okay, great.
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So now we’ve remembered what Heron’s formula is and what the formula for the semiperimeter is.
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We can use this to find the area of our triangle.
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So first of all, we can find our 𝑠, our semiperimeter, cause this is equal to three plus six plus seven over two, which is gonna be equal to 16 over two, which is eight.
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So, we know that the semiperimeter would be eight centimeters.
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So now we’ve got the semiperimeter.
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What we can do is plug this into the Heron’s formula for the area.
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So, what we get is the area is equal to the square root of eight multiplied by eight minus three multiplied by eight minus six multiplied by eight minus seven.
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And because of the way that formula’s set up, it doesn’t matter which one of our sides are 𝑎, 𝑏, or 𝑐, which is gonna be equal to the square root of eight multiplied by five multiplied by two multiplied by one, which is gonna give us root 80.
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And then, what we’re going to do is simplify our root 80.
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And what we’re going to get is root 16 multiplied by root five.
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And that’s because we apply one of our radical or surd rules, which is that root 𝑎 multiplied by root 𝑏 is equal to root 𝑎𝑏.
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And therefore, what we can say that the area of the triangle whose side lengths are three centimeters, six centimeters, and seven centimeters is four root five centimeters squared.
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So that was our first example.
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We’re now gonna take a look at the next example which is gonna be similar, just using some slightly different notation.
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𝐴𝐵𝐶 is a triangle, where 𝐵𝐶 equals 28 centimeters, 𝐴𝐶 equals 20 centimeters, and 𝐴𝐵 equals 24 centimeters.
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Find the area of 𝐴𝐵𝐶 giving the answer to the nearest square centimeter.
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So, the first thing we’ve done is just drawn a quick sketch to visualize what’s happening.
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So, what we have is a triangle.
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And in that triangle, we have three sides, and we know each of their lengths.
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So, therefore, if we know the lengths of each of the sides of our triangle and we want to find the area, then what we’re going to do is use Heron’s formula.
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Well, if we quickly remind ourselves of Heron’s formula, if we have a triangle lengths 𝑎, 𝑏, and 𝑐, then the area of this triangle is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑠 is the semiperimeter, so half of the perimeter of our triangle.
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And we could find that by adding together each of the sides, so 𝑎 plus 𝑏 plus 𝑐, and then dividing it by two.
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So, the first thing we’re gonna do is find out 𝑠.
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And to find this, what we’re gonna do is add together the three side lengths, so 28 plus 20 plus 24, and then divide this by two.
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And this is gonna give us a semiperimeter of 36 centimeters.
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So, therefore, the area is gonna be equal to the square root of 36 multiplied by 36 minus 28 multiplied by 36 minus 20 multiplied by 36 minus 24, which is gonna be equal to the square root of 55296.
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Well, this is equal to 235.1510153.
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However, is this the final answer?
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Well, no because if we look back at the question, we can see that we want the answer to the nearest square centimeter.
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So, therefore, we can say that to the nearest square centimeter, the area of the triangle is 235 centimeters squared.
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So far, what we’ve seen is a couple of examples of how to use Heron’s formula to find the area of a triangle.
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Now, what we’re gonna take a look at is example where we’re gonna use Heron’s formula to find the area of a rhombus.
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The perimeter of the given rhombus is 292 centimeters and the length of 𝐴𝐶 is 116 centimeters.
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Use Heron’s formula to calculate the area of the rhombus, giving the answer to three decimal places.
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Well, the first thing we need to do is remind ourselves of Heron’s formula.
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So, what we’ve got is that if we have a triangle 𝐴𝐵𝐶 and the area is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑠 is the semiperimeter which we can find by adding together each of the sides of the triangle, so 𝑎 plus 𝑏 plus 𝑐, and then dividing it by two.
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So, this is useful.
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Well, we take a look at our diagram and what we’ve got is a rhombus.
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Well, what we can do is if we take a look at the diagonal across our rhombus is see that our rhombus can be divided into two identical triangles.
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However, we still only have one side length of our triangle.
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But what we can do is we can use the perimeter of the rhombus to help us work out what the other side lengths are.
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And that’s because in a rhombus each of the sides are the same length.
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So therefore, if we call each of the side lengths of our rhombus 𝑥, we can say that 𝑥 is gonna be equal to 292 divided by four.
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Well, this is equal to 73.
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So therefore, we could say that each of the side lengths of our rhombus is 73 centimeters.
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And more importantly, if we look at the orange triangle, we’ve now got a triangle where we know all three sides, 116, 73, and 73.
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Well, then, the first thing we want to do is work out what the semiperimeter is.
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And we do that by adding together our side lengths, so 116 plus 73 plus 73, and then dividing it by two.
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And this gives us a semiperimeter of 131 centimeters.
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Okay, great.
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So now what we can do is substitute this and our side lengths into our formula, so our Heron’s formula.
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And when we do that, we get that the area is equal to the square root of 131 multiplied by 131 minus 116 multiplied by 131 minus 73 multiplied by 131 minus 73, which is equal to 2571.0425 et cetera.
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We might think, “Wow, great!
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We found the area.
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Can we finish at this point?”
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But no.
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And why is that?
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Well, if we take a look back at our rhombus, we can see that, in fact, it’s two identical triangles.
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So, therefore, we need to multiply this by two.
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And when we do that, we get 5142.08518 et cetera.
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And then if we look back at the question for the degree of accuracy we want the answer left in, we can see we want it to three decimal places.
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So, therefore, after rounding, we can say that the area of the rhombus is 5142.085 centimeters squared, and we set that to three decimal places.
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So great, we’ve looked at a selection of questions so far.
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We’ve got two more examples now.
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The first one is gonna have a look at a composite figure, so more complex composite figure.
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And we’ll be using Heron’s formula to find the area of it, and that hits one of the objectives of the lesson.
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And then the final problem is a problem-solving question, where we look to find the radius of a circle.
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Find the area of the figure below using Heron’s formula, giving the answer to three decimal places.
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So, the first thing we do is we remind ourselves about Heron’s formula.
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And Heron’s formula tells us what the area is of a triangle if we know all three side lengths.
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So, for instance, if we’ve got the triangle 𝑎𝑏𝑐, then the area is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑠 is a semiperimeter which we can find by adding together the all three side lengths, 𝑎 plus 𝑏 plus 𝑐, and then dividing it by two.
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Well, if we take a look at our shape, it’s a composite made up of two triangles, triangle 𝐴 and triangle 𝐵.
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So, in fact, what we’re gonna do is we’re gonna start with triangle 𝐵.
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And that’s because for triangle 𝐵, we know all three sides.
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So, the first thing we can do is find the semiperimeter of triangle 𝐵.
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And that’s gonna be equal to all three side lengths added together divided by two, so 20 plus 23 plus 16 over two, which is gonna give us a semiperimeter of 29.5 centimeters.
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Okay, great.
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So, now, what we can do is substitute this into Heron’s formula to find the area of triangle 𝐵.
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So, therefore, the area of the triangle is gonna be equal to the square root of 29.5 multiplied by 29.5 minus 20 multiplied by 29.5 minus 23 multiplied by 29.5 minus 16, which is gonna be equal to 156.818166 et cetera centimeters square.
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Okay, great.
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We don’t round at this point because we don’t want any rounding errors before we get to the final answer.
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So, that’s the area of triangle 𝐵.
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Now, let’s move on to triangle 𝐴.
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Well, for triangle 𝐴, we only have two side lengths, and what we need to do is find the other side length and that’s so that we can find the area of the triangle.
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I’m gonna call the other side length 𝑥.
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And because we’ve got a right triangle, what we can do is use the Pythagorean theorem.
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And what the Pythagorean theorem tells us is that if we have a right triangle, then we have 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑐 is the hypotenuse or longest side.
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But what we’re gonna do is rearrange this because what we’re trying to find is the shorter side.
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So, what we’re gonna get is that 𝑥 squared is equal to 20 squared minus 16 squared.
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So, 𝑥 is gonna be equal to root 144.
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So, therefore, the length of 𝑥 is gonna be 12 centimeters.
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So, what we now know is all three lengths of our triangle.
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But what we could do is use Heron’s formula.
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However, because of the type of triangle we have, what we’re gonna instead use is the area of a triangle is equal to a half the base times the height cause we know the base and we know the perpendicular height.
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So, therefore, the area of triangle 𝐴 is gonna be equal to a half multiplied by 12 multiplied by 16, which is equal to 96 centimeters squared.
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So, now, the final stage to find the total area is gonna be to add this to the area of triangle 𝐵.
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Well, when we do that, we’re gonna get 252.8181 et cetera.
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However, the question wants the answer to three decimal places, which is gonna give a final answer of 252.818 centimeters squared to three decimal places.
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Okay, great.
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Now, we’re gonna move on to our final example.
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The lengths of a triangle are 12 centimeters, five centimeters, and 11 centimeters.
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Find the radius of the interior circle touching the sides using the formula 𝑟 equals the area of the triangle 𝐴𝐵𝐶 over 𝑝, where 𝑝 is half of the triangle’s perimeter.
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Well, to be able for us to use the formula for the radius of the circle, what we need to do is find the area of the triangle.
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And what we’re told are three side lengths of our triangle: 12, five, and 11.
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Well, if we have three side lengths of a triangle, then what we can use is Heron’s formula to find the area.
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And Heron’s formula tells us that the area is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑎, 𝑏, and 𝑐 are the side lengths of our triangle.
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And 𝑠 is equal to 𝑎 plus 𝑏 plus 𝑐 over two because it’s the semiperimeter of our triangle.
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Well, actually, this ties in with the formula we have for the radius because we’re told that the formula for the radius is equal to the area of the triangle divided by 𝑝, where 𝑝 is half of the triangle’s perimeter.
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Well, in fact, this is the same as 𝑠 because they both mean the semiperimeter or half of the triangle’s perimeter.
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So, the first thing we want to do is find out the area of the triangle.
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And to do that, firstly, we need to find the semiperimeter or 𝑝, the half-perimeter of the triangle.
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Well, this is gonna be equal to 12 plus five plus 11 over two.
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So, this is gonna be equal to 14 centimeters.
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So great, what we can do now is substitute this into Heron’s formula.
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And when we do that, we’re gonna get the area is equal to the square root of 14 multiplied by 14 minus 12 multiplied by 14 minus five multiplied by 14 minus 11, which is equal to root 756.
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And if we simplify this, we get six root 21.
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And we’re gonna keep it in surd form to keep accuracy, and the units for this will be centimeters squared because it’s our area.
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Okay, great.
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So, now, we have everything we need to substitute into the formula to find the radius.
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What we’ve done is that we sketched what we’re trying to find because what we’re trying to find is the radius of the interior circle which touches the sides of our triangle.
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So, we can say that the radius is equal to six root 21 over 14.
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And we know that because 14 was our 𝑠, our semiperimeter.
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And we’d already said that this is the same as 𝑝.
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So, therefore, this is gonna give us our final answer, which is the radius of the interior circle is three over seven multiplied by root 21 centimeters.
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Okay, great, we looked at a range of different examples covering all the different objectives of the lesson.
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But what we’re gonna do now is just have a look at the key points once again.
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Well, the first and main key point is what Heron’s formula is.
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Heron’s formula allows us to find the area of a triangle when we know all three side lengths.
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And we get that with the formula 𝐴 is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑠 is the semiperimeter.
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So that is half of the perimeter of the triangle.
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And we get that by adding the sides together, so 𝑎 plus 𝑏 plus 𝑐, and then dividing it by two.
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And what we also found in the lesson was an interesting adaptation of Heron’s formula, and that was to find the radius of an interior circle which touches the sides.
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And that formula was that 𝑟 is equal to the area of the triangle, which we could find using Heron’s formula, divided by 𝑝, or in fact 𝑠 because it’s the semiperimeter or half-perimeter of the triangle.