Video Transcript
In this video, we’re going to look
at the triangle inequality which tells us about an important relationship that must
exist between the lengths of the sides in a triangle.
So this is what the triangle
inequality says. It says: The sum of the lengths of
any two sides of a triangle must be greater than the length of the third side. So let’s have a look at this
triangle that I’ve drawn here. What it means is if I take two
sides, so let’s take 𝐴𝐵 and 𝐵𝐶, the sum, so I’m gonna add them together, so the
sum of those two sides 𝐴𝐵 and 𝐵𝐶 must be greater, so greater, than the length of
the third side. So it must be greater than, in this
case, 𝐴𝐶. So that’s one formulation of the
triangle inequality for this triangle.
But it says the sum of the lengths
of any two sides, which means I can also write this inequality down using the other
pairs. So it must also be true that 𝐴𝐵
plus 𝐴𝐶 is greater than 𝐵𝐶. And third, it must also be true
that 𝐵𝐶 plus 𝐴𝐶 is greater than 𝐴𝐵. So whichever pair of sides I
choose, the sum of those two sides must be greater than the length of the third
side.
Now just have a think about why
that is the case, that this inequality has to be true. Suppose I asked you to construct a
triangle with sides of seven centimeters, four centimeters, and two centimeters. Now to construct means to do so
accurately using mathematical equipment, such as a ruler and a pair of
compasses. So what you would do if you were
attempting to construct this triangle. Well, I would start off by
accurately drawing out the base side of seven centimeters. And then the way that you would
draw accurately this side of four centimeters is you would get your pair of
compasses. You would set it to four
centimeters. You would place the point at one
end of this seven-centimeter line. And then you would draw an arc of
all the points four centimeters away from that point there. In order to construct the
two-centimeter side, you do the same thing. You’d set your pair of compasses to
two centimeters, place the point at this end here, and then you draw an arc of all
the points two centimeters away.
Now what you notice is that those
arcs don’t cross at any point. So if you were to try and draw your
triangle, you would end up with a gap where those lines can’t possibly meet to form
the third corner. Now this is why it’s important that
the triangle inequality holds true, because all you can see this triangle didn’t
work, left a gap. And if you look at those
measurements of four, two, and seven, you’ll see that the triangle inequality
doesn’t work because I have that four plus two, which is six, well that isn’t
greater than seven. And that’s why I wasn’t able to
draw this triangle. So let’s look at a question on
this.
The first question says: Is it
possible to form a triangle with side lengths three inches, five inches, and
seven inches?
So what we need to check is
does the triangle inequality hold true for all the different pairs of sides
here. So if you recall, we’ve got
three different inequalities that we need to check. So here’s the first one. If I take the three and the
five, is three plus five greater than seven? And of course, it is. Eight is greater than
seven. So that one works. Now let’s look at the second
pair. So let’s take the five and the
seven. So is the sum of those two
sides greater than the third side? Is five plus seven greater than
three? And of course, 12 is greater
than three. So that one also works.
The third one, we need to take
the final pair of sides. So we need to look at the sum
of the three and the seven. So we’re asking, is three plus
seven greater than five? And yes, 10 is greater than
five. So all three triangle
inequalities are satisfied, which means the answer to the question “is it
possible to form this triangle?”, yes, it is. We do of course need the
working out in order to back up that answer.
Okay, a very similar question,
is it possible to form a triangle with side lengths six meters, seven meters,
and 18 meters?
So we know what we need to do
from the previous one. We need to check whether these
three triangle inequalities hold. But just by casting an eye over
these three measurements, six meters, seven meters, and 18 meters, I can see
already that there’s one that’s not going to work. If I look at the sum of the six
and the seven, so six plus seven is 13. And thirteen is not greater
than 18. So the sum of this pair of
sides does not exceed the length of the third side, which means that, for this
question, no, it isn’t possible to form such a triangle.
Now an important point to note
here, when the answer was yes, we had to check all three of the inequalities and
make sure all three of them hold true. When the answer was no, it’s
enough to demonstrate that just one pair of sides doesn’t satisfy this
inequality. So as long as you can find one
of them that doesn’t work, then it won’t be possible to form the triangle. You don’t actually need to
check the other two.
Okay, the next question says:
Two sides of a triangle are five centimeters and eight centimeters. What is the range of values for
the third side?
So we aren’t asked to find the
third side explicitly. We’re asked to find the range
of possible values that this third side could be. So I don’t know this side,
which means I’m gonna start off by allocating a letter. So I’m gonna call it 𝑥. So I’m starting my working out
with “Let the length of the third side be 𝑥 centimeters.” Now I need to think about those
three triangle inequalities that we saw previously. And to start off with, I’m
gonna think about the fact that if I take this third side 𝑥 and the side of
five centimeters, then the sum of those two sides need to be greater than the
eight centimeters.
So this gives me my first
inequality. And it’s that 𝑥 plus five must
be greater than eight. Now to solve this inequality, I
need to subtract five from both sides. And this tells me that 𝑥 must
be greater than three. So this is my first piece of
information about 𝑥. It has to be bigger than
three.
Now, let’s write down our
second inequality. And it’s gonna be that if I
take the third side 𝑥, an eight centimeters, that has to be bigger than
five. So this gives me the inequality
𝑥 plus eight must be greater than five. Now to solve this, I need to
subtract eight from both sides. And this tells me that 𝑥 must
be greater than negative three. Now that doesn’t really add any
extra information because 𝑥 is representing a length. So by definition, it has to be
a positive value, which means it must be greater than negative three. So we haven’t really gained
anything by using that inequality there.
The third inequality, I need to
look at the five and the eight. And the sum of those must be
greater than 𝑥. So I have five plus eight is
greater than 𝑥. Simplifying that, it tells me
that 13 is greater than 𝑥. Or if I just write that
inequality the other way around, then I have that 𝑥 is less than 13.
Okay, finally, I need to put
all of this information together. So that middle inequality,
remember, didn’t really tell me anything useful. But the other two do. They tell me, in one case, that
𝑥 is greater than three. And then that secondly, it
tells me that 𝑥 is less than 13. So if I pull those two
inequalities together, it gives me a double-sided inequality for 𝑥. So this tells me that 𝑥 is
greater than three but less than 13. And so, that is the range of
possible values for this third side of the triangle.
Now just as an aside here, if
the question had asked for the least integer value of 𝑥, for example, then it
wouldn’t be three. It would be four, because it
has to be strictly greater than three. So we’re looking at the next
integer up after three. So in this question, we formed
three inequalities by looking at the three different pairs of sides of the
triangle, solved the inequalities, and then pulled all that information together
at the end of the question, in order to work out the range of possible values
for this third side.
Okay, the final question, we’re
given a diagram and we’re told: Given that 𝑥 is an integer, so a whole number,
determine the possible values of 𝑥.
And looking at the diagram, we
can see that all three sides are expressed in terms of this letter 𝑥. So just like in the previous
example, we have three inequalities that we need to write down. So the first one, I’m gonna
take the red side and the blue side and sum them together. And when I do that, the
triangle inequality tells me that that must be greater than the third side, the
five 𝑥 minus three. So I have this inequality
here.
Now what I need to do is I need
to solve that inequality. So if I simplify the left-hand
side, I’ve got three 𝑥 plus 𝑥 is four 𝑥 and negative one plus three is
two. So I have four 𝑥 plus two is
greater than five 𝑥 minus three. Next step to solving this
inequality, I want to add three to both sides, which gives me four 𝑥 plus five
is greater than five 𝑥. And then finally, subtract four
𝑥 from both sides, which gives me five is greater than 𝑥. Or if I just turn it around the
other way, I have 𝑥 is less than five. So that gives me my first piece
of information about 𝑥. It’s that it must be less than
five.
Now I need to do the same thing
with the other pairs of sides of this triangle. So if I look at the blue side
and the green side next, and the sum of those two, so then I have that 𝑥 plus
three plus five 𝑥 minus three must be greater than that third side, which is
three 𝑥 minus one. Now at the same time, I’m gonna
write down my third inequality. So this third one is where I
sum the red and the green side. And that must be greater than
the blue side. So that gives me this third
inequality: three 𝑥 minus one plus five 𝑥 minus three is greater than 𝑥 plus
three.
Now I’m not going to go through
the step-by-step solution of both of those inequalities. I have written on the
screen. So if you want to pause the
video and go through it yourself, you can. But it gives us these values
here. 𝑥 is greater than negative a
third, for the first one, which doesn’t really give us any new information
because the third inequality tells us that 𝑥 has to be greater than one. So if 𝑥 has to be greater than
one, it definitely has to be greater than negative a third. And we also need to make sure
that the length of these sides is positive.
So it’s the first and the last
inequalities that give us the most information. 𝑥 has to be greater than one,
but it has to be less than five. So I could write that as a
double-sided inequality, as I did before. But this question actually says
that 𝑥 is an integer. So 𝑥 is an integer somewhere
greater than one but less than five, which means there are three possible
integers that 𝑥 could be. 𝑥 could be two, three, or
four. So that gives me my answer to
this question.
So to summarize then, we’ve seen
what the triangle inequality is and what it tells us about the relationship that
must exist between the three sides of a triangle. We’ve seen how to apply it to
determine whether it is or isn’t possible to create a particular triangle, given
three lengths. And then, we’ve seen how to apply
it to some more complex questions where we’ve had to set up and solve some algebraic
inequalities.
