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The perimeter of an isosceles triangle is 18 centimetres.
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Find the different possible lengths of its sides given they are integers.
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So, let’s say we have our isosceles triangle.
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As it is isosceles, this means that two of the lengths of the triangle will be the same size.
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We’re told that the perimeter of this triangle is 18 centimetres, so this means that sum of the lengths is 18 centimetres.
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So, let’s create the value 𝑥 which is the length of one of our sides.
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Since it’s isosceles, we know that there will also be another length on this triangle that is 𝑥 as well.
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As the perimeter of our triangle is 18, then we can write that the final length will be equal to 18 minus two 𝑥.
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The question asks us to find the different possible lengths of the sides.
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So, we could try different possible values for 𝑥.
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For example, if 𝑥 was seven, then we know that we would have lengths of seven, seven, and four.
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And then, we could try another value of 𝑥.
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However, we can also notice that there’s a very important relationship between 𝑥 and 18 minus two 𝑥.
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And that is that as our value of 𝑥 increases, 18 minus two 𝑥 decreases.
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And it does so in a linear way, since an increase in 𝑥 results in a corresponding decrease in 18 minus two 𝑥.
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If we have two variables in a linear relationship, then we can plot a graph to show it.
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So, let’s draw a graph with our 𝑥-value on the 𝑥-axis and our 18 minus two 𝑥-values on our 𝑦-axis.
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We can use a table of values for our 𝑥-values and their corresponding 18 minus two 𝑥-values.
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So, let’s start with 𝑥 equal zero.
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In this case, 18 minus two 𝑥 would be equal to 18 minus zero, which is 18.
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When 𝑥 equals one, 18 minus two 𝑥 would be 18 minus two, giving us 16.
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Next, when 𝑥 equals two, 18 minus two 𝑥 would be 14.
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And we can continue choosing values of 𝑥 up till 𝑥 equals nine.
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And 18 minus two 𝑥 would be zero.
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Notice that if we did try a value of 𝑥 equals 10, this would give us a corresponding value for 18 minus two 𝑥 of negative two.
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And in the context of our question, this means that one of the side lengths of the triangle would be negative two, which isn’t valid.
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So, let’s now use the values we’ve worked out to draw a graph.
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So, here, we have our graph showing very clearly the linear relationship between 𝑥 and 18 minus two 𝑥.
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So, let’s now use this graph to find the different possible lengths of the sides of the triangle.
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We can see that, for example, the coordinate one, 16 would mean that our triangle had two lengths of one and a length of 16.
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The point two, 14 on our graph would mean two lengths of two and one length of 14.
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And then, we could continue to list all the different possible values for the lengths of the sides.
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However, when we look at these options, we need to apply some common sense and some mathematical reasoning.
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If we look at our option of zero, zero, 18 and nine, nine, zero, both of these would include sides of zero centimetres.
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So, we can eliminate both of these possibilities.
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Let’s take a look at our second listing option of one, one, and 16.
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If we were to attempt to draw this triangle and placed a line of 16 units long, then no matter how we drew our other two lengths of unit one, we could never form a triangle.
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So, we can eliminate this possible option.
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For the same reason, we can also eliminate our options of two, two, 14; three, three, 12; and four, four, 10.
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And now, if we take a look at the remaining options for our side lengths, we can see one set that’s different to the others.
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The triangle that has three side lengths of six, this would be an equilateral triangle and not an isosceles triangle that we’re looking for.
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So, it also can be eliminated.
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This leaves us with three remaining possible options.
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So, we can say that the different options for the lengths of the sides in the triangle are eight, eight, and two; seven, seven, and four; or five, five, and eight.