The perimeter of an isosceles triangle is 18 centimetres. Find the different possible lengths of its sides given they are integers.
So, let’s say we have our isosceles triangle. As it is isosceles, this means that two of the lengths of the triangle will be the same size. We’re told that the perimeter of this triangle is 18 centimetres, so this means that sum of the lengths is 18 centimetres. So, let’s create the value 𝑥 which is the length of one of our sides. Since it’s isosceles, we know that there will also be another length on this triangle that is 𝑥 as well. As the perimeter of our triangle is 18, then we can write that the final length will be equal to 18 minus two 𝑥.
The question asks us to find the different possible lengths of the sides. So, we could try different possible values for 𝑥. For example, if 𝑥 was seven, then we know that we would have lengths of seven, seven, and four. And then, we could try another value of 𝑥. However, we can also notice that there’s a very important relationship between 𝑥 and 18 minus two 𝑥. And that is that as our value of 𝑥 increases, 18 minus two 𝑥 decreases. And it does so in a linear way, since an increase in 𝑥 results in a corresponding decrease in 18 minus two 𝑥.
If we have two variables in a linear relationship, then we can plot a graph to show it. 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. We can use a table of values for our 𝑥-values and their corresponding 18 minus two 𝑥-values. So, let’s start with 𝑥 equal zero. In this case, 18 minus two 𝑥 would be equal to 18 minus zero, which is 18. When 𝑥 equals one, 18 minus two 𝑥 would be 18 minus two, giving us 16.
Next, when 𝑥 equals two, 18 minus two 𝑥 would be 14. And we can continue choosing values of 𝑥 up till 𝑥 equals nine. And 18 minus two 𝑥 would be zero. 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. 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.
So, let’s now use the values we’ve worked out to draw a graph. So, here, we have our graph showing very clearly the linear relationship between 𝑥 and 18 minus two 𝑥. So, let’s now use this graph to find the different possible lengths of the sides of the triangle. 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. The point two, 14 on our graph would mean two lengths of two and one length of 14. And then, we could continue to list all the different possible values for the lengths of the sides.
However, when we look at these options, we need to apply some common sense and some mathematical reasoning. If we look at our option of zero, zero, 18 and nine, nine, zero, both of these would include sides of zero centimetres. So, we can eliminate both of these possibilities. Let’s take a look at our second listing option of one, one, and 16. 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. So, we can eliminate this possible option.
For the same reason, we can also eliminate our options of two, two, 14; three, three, 12; and four, four, 10. 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. 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. So, it also can be eliminated. This leaves us with three remaining possible options. 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.