In what interval must the length of 𝐴𝐶 lie?
We’ve been given a diagram of a triangle. We’ve been told the length of two of the sides, but not the length of the third. We haven’t been asked to calculate the length of the third side, but rather to give an interval in which it must lie, so a range of possible values for the length of the third side. This suggests that we’re going to be using the triangle inequality to answer this question.
So let’s recall what the triangle inequality tells us. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For our triangle, this means there are three statements that must be true. In each case, we’ve taken two of the sides and sum them and then set that it must be greater than the third side. What we’re going to do then, is substitute the values of the two sides we know, so that’s 𝐴𝐵 and 𝐵𝐶, into each of these inequalities. We’ll then rearrange them to see what information this gives us about 𝐴𝐶.
Let’s begin then with the first inequality on the left of the screen. Substituting the value of 16 for 𝐴𝐵 and 19.3 for 𝐵𝐶, we have that 16 plus 19.3 is greater than 𝐴𝐶. This simplifies to 35.3 is greater than 𝐴𝐶. Or, if you write this inequality the other way around, we have that 𝐴𝐶 must be less than 35.3.
So this is perhaps part of our answer. We now have an upper bound on 𝐴𝐶. We know it can’t be greater than 35.3. Now let’s look at each of the other two inequalities. Substituting the values of 𝐴𝐵 and 𝐵𝐶 into the second inequality, gives us 16 plus 𝐴𝐶 is greater than 19.3. In order to solve this inequality for 𝐴𝐶, we need to subtract 16 from each side. This tells us that 𝐴𝐶 is greater than 3.3. So now we have a lower bound on the length of 𝐴𝐶.
Let’s look at the third inequality and see if it gives us any further information about 𝐴𝐶. Substituting the length of 𝐴𝐵 and 𝐵𝐶 again, tells us that 19.3 plus 𝐴𝐶 must be greater than 16. Now in order to solve for 𝐴𝐶, we need to subtract 19.3 from each side. And this tells us that 𝐴𝐶 is greater than negative 3.3. Now that has to be true because 𝐴𝐶 represents the length of one side of a triangle, so it has to take a positive value. So whilst this inequality is true, it hasn’t given us any meaningful information about the length of 𝐴𝐶, which means it’s just the first two inequalities that we need to look at.
So the first two inequalities told us that 𝐴𝐶 had to be greater than 3.3 and had to be less than 35.3. The question has asked us for an interval, so we need to give our answer using interval notation. So in interval notation, we list the lower bound of the interval, 3.3, followed by the upper bound of the interval, 35.3.
And now we need to consider what type of brackets go around this interval. These, remember, are strict inequalities as the triangle inequality states that the sum must be greater, not greater than or equal to. This means that 𝐴𝐶 can’t actually take the values of 3.3 or 35.3. It must be strictly greater than and strictly less than these two values. Therefore, we need to use the notation for an open interval. So we’ll use brackets such as these. You may also see an open interval referred to using square brackets but facing outwards to indicate that the two endpoints of the interval are not included.
So the interval of values in which the length of 𝐴𝐶 must lie is between 3.3 and 35.3. But this is an open interval, so the endpoints of that interval are not included.