Welcome to a video on the pythagorean theorem and its converse
to determine if the triangle is an acute triangle, a right triangle
or an obtuse triangle. We'll also take a look at the triangle inequality
to determine if the triangle is even possible, given the length
three segments. Let's start by reviewing the pythagorean theorem,
that states that c squared is equal to a squared, plus b squared,
where c would be the hypotenuse, or the longest side
as we see here. And a and b would be the lengths
of the two legs. So if we know the length of these three sides,
and c squared is equal to a squared, plus b squared,
we would have a right triangle. But we can also use the relationship between c
squared,
and the sum of a squared and b squared, to determine if we have an acute triangle,
or an obtuse triangle. Meaning if c squared
is less than a squared, plus b squared,
then we would have an acute triangle. However, to apply this we do have to
remember that c will always be the longest side.
So in this case it looks like this would be c,
and these two sides would be a and b. And the last case is,
if c squared is greater than a squared, plus b squared,
then if this is the case, then we have an obtuse triangle.
Meaning that one of the angles would be between ninety and
one hundred eighty degrees. Again, it looks like this would be side c,
and this would be a and b. Before we take a look at some examples of this,
we do need to talk about something called the triangle inequality.
When we're given the length of three segments, it's not always possible to
form a triangle.
In order to construct a triangle, the sum of the two shorter sides
must be greater than the longest side. And that should make sense because if this
was side c,
and this was side a and side b. If the sum of
a and b are not greater than c,
these two sides could not reach each other, and therefore could not form a
triangle.
So we do want to make sure that we always check to make sure that
a plus b is greater than c. To make sure that we can even form a triangle.
Again this would be c. Here we'd have and and b.
Okay, let's look at our examples. We want to determine if the lengths represent the
sides of an acute,
right, or obtuse triangle, if a triangle is possible.
Let's start off by labeling the longest side c,
the two shorter sides would be a and b. Notice that seven plus twenty-four is greater than
twenty-five,
so we can form a triangle. So now we'll go ahead and determine what's the relationship
between c squared,
and a squared, plus b squared. This will tell us where there's an acute,
right, or obtuse triangle. So we want to know how these two relate.
So we'll leave a little blank there. And then we can fill in whether it's
less than, equal to, or greater than, when we finish.
Since c is twenty-five, we'll have twenty-five squared.
And on the right we'll have seven squared, plus twenty-four squared.
Well, twenty-five squared is six hundred twenty-five.
Seven squared is forty-nine,
and twenty-four squared is five hundred seventy-six.
Well in this case we have six hundred twenty-five on the left,
we also have six hundred twenty-five on the right.
So this tells us that c squared is equal to
a squared plus b squared, therefore we have a
right triangle. Okay on number two,
this would be side c,
let's go then call this a, and this b. Notice that a plus b,
or five plus twelve is greater than sixteen, so we can form a triangle.
Let's go ahead and set up the relationship between c squared,
and a squared, plus b squared again. So now we'll have sixteen squared on the left.
Then we'll have five squared plus twelve squared on the right.
So we'll have two hundred fifty-six. On
the right we'll have twenty-five plus one hundred forty-four.
So we'll have two hundred fifty-six.
This is going to give us what? One hundred sixty-nine.
So what we've discovered is that c squared is greater than
a squared, plus b squared. And therefore this tells us we have an obtuse
triangle.
Meaning one of the angles will be an obtuse angle, or an angle between ninety,
and one hundred eighty degrees.
Let's go ahead and take a look at two more.
Again, the process is going to be the same. C is the longest side,
a and b are the two shorter sides. Notice that a plus b
is greater than nine, so we can form a triangle.
So again let's set it up. C squared on the left,
a squared, plus b squared on the right.
So we'll have nine squared, and then on the right we'll have six
squared, plus eight squared. So we'll have
eighty-one, thirty-six,
and sixty-four. And on the right side,
thirty-six plus sixty-four, that's a hundred. So now we have c squared is
less than a squared, plus b squared which means we have an acute
triangle. All angles of this triangle
will be less than ninety degrees.
And let's finish with one more. This would be c.
This would be a. This would be b. Again you can interchange a and b
but c has to be the longest side. Notice in this problem that three plus
five
is not greater than nine,
so this tells us that no triangle is even possible.
So it is important that we check the triangle inequality.
So if we applied the pythagorean theorem and it's converse to
these three lengths, it would show that it's an obtuse triangle,
but in fact it's not because a triangle is not even possible.
Okay. I hope you found this video helpful. Thank you for watching.