### Video Transcript

So in this lesson, what weโre gonna
look at is the converse of the Pythagorean theorem. But what is the converse of the
Pythagorean theorem?

Well, the converse of the
Pythagorean theorem is where we use the Pythagorean theorem to prove whether a
triangle is a right triangle or not. Well, first of all, we need to
remind ourselves what the Pythagorean theorem is. And the Pythagorean theorem is ๐
squared plus ๐ squared equals ๐ squared, where ๐ is the hypotenuse, which is the
longest side opposite the right angle. And ๐ and ๐ are the other two
sides. And it doesnโt matter which one is
which. And itโs worth noting that the
Pythagorean theorem only works in a right or right-angled triangle.

So, what does this mean? Well, it means that the sum of the
areas of the squares on the two shorter sides or legs is equal to the area of the
square formed on the longest side, the hypotenuse. So, therefore, if we have a
triangle where this doesnโt work, so ๐ squared plus ๐ squared isnโt equal to ๐
squared, then we can conclude that it wonโt be a right triangle.

Okay, great. So, now, we know what it is and how
weโre going to use it. Letโs have a little look at how it
all works.

The Pythagorean theorem states
that, in a right triangle, the area of a square on the hypotenuse is equal to the
sum of the areas of the squares on the legs. Does this mean that a triangle
where ๐ squared equals ๐ squared plus ๐ squared is necessarily a right
triangle? Let us assume that triangle ๐ด๐ต๐ถ
is of side lengths ๐, ๐, and ๐ with ๐ squared equals ๐ squared plus ๐
squared. Let triangle ๐ท๐ต๐ถ be a right
triangle of side lengths ๐, ๐, and ๐. Part (1) Using the Pythagorean
theorem, what can you say about the relationship between ๐, ๐, and ๐?

In this problem, we have the
question, does this mean that a triangle where ๐ squared equals ๐ squared plus ๐
squared is necessarily a right triangle? And what weโre gonna do is weโre
gonna answer this question through each of the stages of this problem. Well, as the question states, the
Pythagorean theorem says that ๐ squared equals ๐ squared plus ๐ squared. Well, this is where ๐ is the
hypotenuse. And the hypotenuse is the longest
side opposite the right angle.

Well, in this orange side here,
itโs ๐ is our hypotenuse because itโs opposite the right angle and itโs the longest
side. So, therefore, as weโre told that
triangle ๐ท๐ต๐ถ is a right triangle, this can give us a relationship. And that is that ๐ squared is
gonna be equal to ๐ squared plus ๐ squared, which means weโve answered the first
part of the question.

So, now, letโs move on to the
second part of the question.

We know that for triangle ๐ด๐ต๐ถ,
๐ squared equals ๐ squared plus ๐ squared. What do you conclude about ๐?

Well, if weโve got ๐ squared
equals ๐ squared plus ๐ squared, what we can do is we can use some
substitution. Because what we can do is we can
sub in ๐ squared for ๐ squared plus ๐ squared cause we know that ๐ squared
equals ๐ squared plus ๐ squared. So, therefore, what weโre gonna
have is ๐ squared is equal to ๐ squared. So, now, we can form a relationship
between ๐ and ๐. And thatโs if we take the square
root of both sides of the equation, what weโre gonna be left with is ๐ is equal to
๐. And we can ignore the negative
values because weโre looking at lengths. Well, therefore, weโve answered
this part of the question cause weโve concluded about ๐ that ๐ is equal to ๐.

Okay, so, letโs move on to the next
part.

Well, the next part, of the
question states, is it possible to construct different triangles with the
same-length sides?

Well, the answer is no. And thatโs because if two triangles
have the same-length sides, and they are in fact congruent. Thatโs because thatโs one of the
ways that we use to prove congruency. Because itโs called SSS, and it
means side-side-side. And if two things are congruent,
what this means is that they are exactly the same. So, weโve answered this part of the
question.

Letโs move on to the next part.

So, for the final part of the
question, what do you conclude about triangle ๐ด๐ต๐ถ?

Well, we know that triangle ๐ท๐ต๐ถ
and triangle ๐ด๐ต๐ถ both have the same length ๐. They both have the same length
๐. And weโve already stated that ๐ is
equal to ๐. So, therefore, first of all, we
must say that itโs congruent to triangle ๐ท๐ต๐ถ. And thatโs because all the sides
are the same. So, therefore, if theyโre
congruent, what we can say is that triangle ๐ด๐ต๐ถ has a right angle at ๐ถ.

So, what weโve done here is weโve
shown one way of demonstrating the Pythagorean theorem. And weโve also answered the
question, does this mean that a triangle where ๐ squared equals ๐ squared plus ๐
squared is necessarily a right triangle? And the answer is yes, it is. And weโve shown that with each of
our steps. So, therefore, our full conclusion
about triangle ๐ด๐ต๐ถ is that itโs congruent to triangle ๐ท๐ต๐ถ. So, it has a right angle at ๐ถ.

So, great, weโve now shown the
Pythagorean theorem. But what we wanna do now is see how
we can use it to prove whether or not a triangle is, in fact, right angled. So, what weโre gonna do in this
problem is weโre gonna use the converse of the Pythagorean theorem to show whether a
triangle could in fact be a right-angled or right triangle.

Can the lengths 7.9 centimeters,
8.1 centimeters, and 5.3 centimeters form a right-angled triangle?

Well, the first thing we could say
about these three lengths is that we can definitely form a triangle with them. And thatโs because when we add
together the two shorter sides, we get 13.2. And 13.2 is greater than 8.1, which
is the other side. So, therefore, we know we could
form a triangle because if it was less than 8.1, then we couldnโt join them
together. So, we couldnโt form a
triangle. And if it was equal to 8.1, then
itโll just form a straight line. Okay, great. We know we can form a triangle, but
letโs see if we can form a right or right-angled triangle.

So, the first thing weโre gonna do
is recall the Pythagorean theorem. And that tells that ๐ squared
equals ๐ squared plus ๐ squared, where ๐ is our longest side or the
hypotenuse. And so, what it says is that the
sum of the area of the squares on the two shorter sides is equal to the sum of the
area of the square on the longest side. So, therefore, in our question, we
can say that if we square 5.3 and 7.9 and add them together, itโs got to be equal to
the square of 8.1 if we want it to be a right triangle. And proving it in this way is using
something called the converse of the Pythagorean theorem.

So, the first thing weโre gonna
calculate is 7.9 squared plus 5.3 squared, which gives us 62.41 plus 28.09, which is
equal to 90.5. So, now, what we want to do is we
want to work out the square of the longest side, which is 8.1. Well, we can already estimate to
see that itโs not gonna be the same as 90.5. Because if we think about 8.1
squared, well, itโs almost equal to eight squared. And eight squared is equal to
64. So, therefore, our value is
definitely gonna be lower than 90.5, which it is cause itโs 65.61. So, therefore, we can conclude that
the sum of the squares on the two shorter sides is not gonna be equal to the sum of
the square on the longest side cause 90.5 is not equal to 65.61. So, therefore, we cannot form a
right-angled triangle or a right triangle with these three lengths.

And we couldโve guessed this from
the beginning. Because if we have a look at the
values weโve got, we got 7.9 and 5.3. Well, 7.9 is very close to
eight. So, therefore, if weโre gonna have
eight squared add 5.3 squared is gonna be greater than 8.1 squared because, again,
8.1 squared is very close to eight squared. So that would have been a good
hypothesis that we could have had from the beginning.

Okay, great. So, weโve shown that we can use the
converse of the Pythagorean theorem to show whether itโs a right triangle or
not. But now, what weโre gonna do is
weโre gonna have a look at a question thatโs a bit more involved. So, we can use some more skills
alongside our Pythagorean theorem.

In rectangle ๐ด๐ต๐ถ๐ท, suppose ๐ด๐ธ
equals eight, ๐ท๐ธ equals two, and ๐ท๐ถ equals four. Is triangle ๐ต๐ธ๐ถ right
angled?

Well, the first thing weโre gonna
do in this question is add on the information that we know. Well, firstly, we know that ๐ด๐ธ is
equal to eight. Then, we know that ๐ท๐ธ is equal to
two. And then, weโre told ๐ท๐ถ is equal
to four. Okay, great. Thatโs all the information weโve
been told. But also we can use this to
conclude some more information. Well, first of all, ๐ด๐ต must also
be four. Thatโs cause itโs a rectangle. So, ๐ด๐ต must be equal to ๐ท๐ถ. And ๐ต๐ถ must be equal to two plus
eight because itโs these two values added together cause itโs the same as the
lengths of ๐ด๐ธ and ๐ท๐ธ together. So, this is gonna have a length of
10.

So, weโve now labelled all our
sides. And what we can do is now mark on a
couple of right angles that are gonna prove useful. And we could do that because we
know itโs a rectangle. So, a rectangle would have a right
angle at each of its corners. So, what weโre asked to do in this
question is work out whether triangle ๐ต๐ธ๐ถ is right-angled or a right
triangle. And the way we can do that is by
using the converse of the Pythagorean theorem. And the Pythagorean theorem states
that ๐ squared is equal to ๐ squared plus ๐ squared, where ๐ is our
hypotenuse.

And what we mean by the converse is
that if this isnโt true, then therefore it cannot be a right or right-angled
triangle as the Pythagorean theorem is only true for right-angled triangles. Well, in order to do this and use
this to see whether triangle ๐ต๐ธ๐ถ is, in fact, right angled, well, what weโre
gonna need to do is find out each of the lengths. And as we donโt know the lengths
๐ถ๐ธ or ๐ต๐ธ, weโre gonna have to find out these first.

Well, what weโre gonna do is weโre
gonna start with the side ๐ถ๐ธ. And we can find this out by forming
triangle ๐ถ๐ท๐ธ. And this is a right triangle cause
as weโve shown thereโs a right angle in top left. So, what we can do is we can use
the Pythagorean theorem to find out the length of ๐ถ๐ธ. So, therefore, we can say that ๐ถ๐ธ
squared is gonna be equal to two squared plus four squared. And thatโs because ๐ถ๐ธ is our
hypotenuse because itโs opposite the right angle and the longest side. So, therefore, ๐ถ๐ธ squared is
gonna be equal to four plus 16, which is gonna be equal to 20.

Now, in fact, we donโt have to
actually find out what ๐ถ๐ธ is. Because when weโre gonna work out
the next part, so weโre gonna look at triangle ๐ต๐ธ๐ถ. Weโre gonna have each of the sides
squared. So, we can keep this answer weโve
got here. If we wanted to work out what ๐ถ๐ธ
was, we just take the square root of both sides of this equation here weโve got. And if we did that, we get two
values because we could get a negative or a positive. But we could disregard the negative
value because weโre looking at a length. So, weโre gonna need positive
values.

Okay, great. So, now, we have the information we
need because we know what ๐ถ๐ธ โ well, more importantly, what ๐ถ๐ธ squared โ is
equal to. So, now, letโs move on and find out
๐ต๐ธ. And to do that, what we can do is
look at this right triangle we have on the right-hand side of our rectangle because
weโve got the triangle ๐ด๐ต๐ธ. So, therefore, what we can say is
that ๐ต๐ธ squared is gonna be equal to eight squared plus four squared. So, therefore, ๐ต๐ธ squared is
gonna be equal to 64 plus 16. So, therefore, ๐ต๐ธ squared is
gonna be equal to 80. So, as before, this is all we need
for this problem. However, we couldโve done the same
as the last time and taken the root of both sides to find out what ๐ต๐ธ was.

So, now, what we can do is use all
the information weโve got to work out whether triangle ๐ต๐ธ๐ถ is, in fact, a right
triangle. Because if it is, then ๐ถ๐ธ squared
plus ๐ต๐ธ squared is gonna be equal to ๐ต๐ถ squared because ๐ต๐ถ is the longest
side. Well, first of all, what weโre
gonna do is work out ๐ถ๐ธ squared plus ๐ต๐ธ squared. Well, this is gonna be 20 add 80
cause Iโve shown that we already had ๐ถ๐ธ squared and ๐ต๐ธ squared.

However, if we just have the
lengths root 20 and root 80, we couldโve worked this out because root 20 squared
would be 20 and root 80 squared would just be 80. And thatโs because weโve got a rule
that tells us that if weโve got root ๐ multiplied by root ๐, itโs just equal to
๐. So, root ๐ all squared is just
equal to ๐. So, therefore, we know that ๐ถ๐ธ
squared plus ๐ต๐ธ squared is gonna be equal to 100 cause 20 add 80 is 100.

So, now, what we want to do is we
want to work out what ๐ต๐ถ squared is going to be because if this is the same as
๐ถ๐ธ squared plus ๐ต๐ธ squared, then weโve got a right triangle. Well, the square of the longest
side ๐ต๐ถ is gonna be equal to 10 squared. So, therefore, ๐ต๐ถ squared is
gonna be equal to 100. Well, this is the same as the value
we got when we added the squares of the two shorter sides. So, therefore, what weโve done is
weโve met the criteria for the Pythagorean theorem because ๐ squared plus ๐
squared is in fact equal to ๐ squared. So, therefore, we can say yes, the
triangle ๐ต๐ธ๐ถ is right-angled or a right triangle.

So, in this lesson, weโve covered
various different examples, showing different skills. So, weโve had a look at how the
Pythagorean theorem is formed. Weโve also taken a look at the
converse of the Pythagorean theorem and how it can be used to show whether three
lines can form a right or a right-angled triangle. Weโve also looked at it in this
question in a different format that involved a rectangle and different
triangles.

And next, weโre gonna have a look
at some coordinate geometry.

Two lines intersect at the point
๐ด: three, negative one. One line goes through the point ๐ต:
five, one, and the other goes through the point ๐ถ: negative two, six. Find the lengths of the line
segments ๐ด๐ต, ๐ด๐ถ, and ๐ต๐ถ.

So, what Iโve done first of all to
help us understand what is going on is Iโve drawn a sketch of the three points that
weโve got given. So, to find the lengths of our
three line segments, what weโre gonna use is something called the distance between
points formula. So, what the distance formula
states is that this distance between two points is equal to the square root of ๐ฅ
two minus ๐ฅ one all squared plus ๐ฆ two minus ๐ฆ one all squared. So, itโs the square root of the
change in our ๐ฅ-coordinate squared plus the change in our ๐ฆ-coordinate
squared.

But where does this formula come
from? Well, in fact, itโs an adaptation
of the Pythagorean theorem. Because if weโve got two points ๐ฅ
one, ๐ฆ one and ๐ฅ two, ๐ฆ two, well, the distance between these two points is, in
fact, gonna be the hypotenuse of a right triangle. And thatโs because if we have a
look here, if we form a right triangle, weโd have the change of ๐ฅ would be the
bottom length and the change of ๐ฆ would be our vertical length. So, therefore, our hypotenuse would
be our ๐. So, in that case, if we thought
about the Pythagorean theorem, this states that ๐ squared equals ๐ squared plus ๐
squared. Well, weโd have our ๐ would be our
๐. And then, we could have our ๐ฅ two
minus ๐ฅ one. So, our change in ๐ฅ could be our
๐. And our ๐ฆ two minus ๐ฆ one could
be our ๐.

So, therefore, we can see that in
fact, this would be finding ๐, our distance, using the Pythagorean theorem. Because if we wanted to find out
what ๐ was or ๐ was, it would in fact just be the square root of ๐ squared plus
๐ squared, which is what we had at the top. Brilliant! Okay, now, we know the distance
formula and where itโs come from, letโs find the lengths of the line segments ๐ด๐ต,
๐ด๐ถ, and ๐ต๐ถ.

So, using this, what we can say is
that ๐ด๐ต is gonna be equal to the square root of five minus three all squared plus
one minus negative one all squared, which is gonna be the change in our
๐ฅ-coordinate squared plus the change in our ๐ฆ-coordinate squared. Itโs worth noting that it doesnโt
matter which way round theyโre going to be because either way would give us the same
result because theyโre squared. So, for instance, five minus three
is two. Two squared is four. Three minus five is negative
two. Negative two squared is also
four. This is gonna give us root ๐,
which will simplify to two root two. We did that using a surd
relationship.

So then, if we move on to ๐ด๐ถ,
itโs gonna be equal to the square root of three minus negative two all squared plus
negative one minus six all squared. And this is gonna give us root
74. And then, ๐ต๐ถ can also be found
using the same method and itโs also gonna be root 74.

So now weโve answered those parts,
letโs move on to the next parts of the question.

So, using the Pythagorean theorem,
decide is triangle ๐ด๐ต๐ถ a right triangle. And hence are the two lines
perpendicular?

As I already stated, the
Pythagorean theorem says that ๐ squared equals ๐ squared plus ๐ squared, where ๐
is our longest side, the hypotenuse. Well, if we look at the three
lengths that make up our triangle, we can see that the shortest length must be two
root two. So, therefore, the longest side
must be ๐ด๐ถ or ๐ต๐ถ, but in fact theyโre the same length. So, therefore, we cannot have a
hypotenuse or longest side with this triangle.

So, therefore, we can say that
triangle ๐ด๐ต๐ถ is not a right triangle because the Pythagorean theorem cannot be
met because two root two all squared plus root 74 all squared cannot be equal to
root 74 all squared. And, similarly, the two lines are
not perpendicular to each other because theyโre not at right angles to each other
because there is no right triangle.

So, as weโre at the end of this
lesson, we can look at the key points. First of all, weโve got ๐ squared
equals ๐ squared plus ๐ squared. This is Pythagorean theorem, where
๐ is the hypotenuse. And if the Pythagorean theorem is
not true for a triangle, it cannot be a right triangle. And we know that the converse of
the Pythagorean theorem can be used to determine if a triangle has a right angle or
not. And finally, if there is no longest
side, a triangle cannot be a right triangle. And thatโs because it wouldnโt meet
the conditions that are set out by the Pythagorean theorem, which has to apply to
right triangles only.