Lesson Video: The Converse of the Pythagorean Theorem | Nagwa Lesson Video: The Converse of the Pythagorean Theorem | Nagwa

# Lesson Video: The Converse of the Pythagorean Theorem Mathematics • 8th Grade

In this video, we will learn how to use the converse of the Pythagorean theorem to determine whether a triangle is a right triangle.

17:57

### Video Transcript

So in this lesson, what weโre 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 doesnโt matter which one is which. And itโs 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 doesnโt work, so ๐ squared plus ๐ squared isnโt equal to ๐ squared, then we can conclude that it wonโt be a right triangle.

Okay, great. So, now, we know what it is and how weโre going to use it. Letโs 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 ๐ด๐ต๐ถ is of side lengths ๐, ๐, and ๐ with ๐ squared equals ๐ squared plus ๐ squared. Let triangle ๐ท๐ต๐ถ 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 weโre gonna do is weโre 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, itโs ๐ is our hypotenuse because itโs opposite the right angle and itโs the longest side. So, therefore, as weโre told that triangle ๐ท๐ต๐ถ 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 weโve answered the first part of the question.

So, now, letโs move on to the second part of the question.

We know that for triangle ๐ด๐ต๐ถ, ๐ squared equals ๐ squared plus ๐ squared. What do you conclude about ๐?

Well, if weโve 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 weโre gonna have is ๐ squared is equal to ๐ squared. So, now, we can form a relationship between ๐ and ๐. And thatโs if we take the square root of both sides of the equation, what weโre gonna be left with is ๐ is equal to ๐. And we can ignore the negative values because weโre looking at lengths. Well, therefore, weโve answered this part of the question cause weโve concluded about ๐ that ๐ is equal to ๐.

Okay, so, letโs 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 thatโs because if two triangles have the same-length sides, and they are in fact congruent. Thatโs because thatโs one of the ways that we use to prove congruency. Because itโs 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, weโve answered this part of the question.

Letโs move on to the next part.

So, for the final part of the question, what do you conclude about triangle ๐ด๐ต๐ถ?

Well, we know that triangle ๐ท๐ต๐ถ and triangle ๐ด๐ต๐ถ both have the same length ๐. They both have the same length ๐. And weโve already stated that ๐ is equal to ๐. So, therefore, first of all, we must say that itโs congruent to triangle ๐ท๐ต๐ถ. And thatโs because all the sides are the same. So, therefore, if theyโre congruent, what we can say is that triangle ๐ด๐ต๐ถ has a right angle at ๐ถ.

So, what weโve done here is weโve shown one way of demonstrating the Pythagorean theorem. And weโve 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 weโve shown that with each of our steps. So, therefore, our full conclusion about triangle ๐ด๐ต๐ถ is that itโs congruent to triangle ๐ท๐ต๐ถ. So, it has a right angle at ๐ถ.

So, great, weโve 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 weโre gonna do in this problem is weโre 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 thatโs 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 couldnโt join them together. So, we couldnโt form a triangle. And if it was equal to 8.1, then itโll just form a straight line. Okay, great. We know we can form a triangle, but letโs see if we can form a right or right-angled triangle.

So, the first thing weโre 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, itโs 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 weโre 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 itโs not gonna be the same as 90.5. Because if we think about 8.1 squared, well, itโs 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 itโs 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 couldโve guessed this from the beginning. Because if we have a look at the values weโve got, we got 7.9 and 5.3. Well, 7.9 is very close to eight. So, therefore, if weโre 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, weโve shown that we can use the converse of the Pythagorean theorem to show whether itโs a right triangle or not. But now, what weโre gonna do is weโre gonna have a look at a question thatโs a bit more involved. So, we can use some more skills alongside our Pythagorean theorem.

In rectangle ๐ด๐ต๐ถ๐ท, suppose ๐ด๐ธ equals eight, ๐ท๐ธ equals two, and ๐ท๐ถ equals four. Is triangle ๐ต๐ธ๐ถ right angled?

Well, the first thing weโre gonna do in this question is add on the information that we know. Well, firstly, we know that ๐ด๐ธ is equal to eight. Then, we know that ๐ท๐ธ is equal to two. And then, weโre told ๐ท๐ถ is equal to four. Okay, great. Thatโs all the information weโve been told. But also we can use this to conclude some more information. Well, first of all, ๐ด๐ต must also be four. Thatโs cause itโs a rectangle. So, ๐ด๐ต must be equal to ๐ท๐ถ. And ๐ต๐ถ must be equal to two plus eight because itโs these two values added together cause itโs the same as the lengths of ๐ด๐ธ and ๐ท๐ธ together. So, this is gonna have a length of 10.

So, weโve 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 itโs a rectangle. So, a rectangle would have a right angle at each of its corners. So, what weโre asked to do in this question is work out whether triangle ๐ต๐ธ๐ถ 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 isnโt 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 ๐ต๐ธ๐ถ is, in fact, right angled, well, what weโre gonna need to do is find out each of the lengths. And as we donโt know the lengths ๐ถ๐ธ or ๐ต๐ธ, weโre gonna have to find out these first.

Well, what weโre gonna do is weโre gonna start with the side ๐ถ๐ธ. And we can find this out by forming triangle ๐ถ๐ท๐ธ. And this is a right triangle cause as weโve shown thereโs a right angle in top left. So, what we can do is we can use the Pythagorean theorem to find out the length of ๐ถ๐ธ. So, therefore, we can say that ๐ถ๐ธ squared is gonna be equal to two squared plus four squared. And thatโs because ๐ถ๐ธ is our hypotenuse because itโs opposite the right angle and the longest side. So, therefore, ๐ถ๐ธ squared is gonna be equal to four plus 16, which is gonna be equal to 20.

Now, in fact, we donโt have to actually find out what ๐ถ๐ธ is. Because when weโre gonna work out the next part, so weโre gonna look at triangle ๐ต๐ธ๐ถ. Weโre gonna have each of the sides squared. So, we can keep this answer weโve got here. If we wanted to work out what ๐ถ๐ธ was, we just take the square root of both sides of this equation here weโve 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 weโre looking at a length. So, weโre gonna need positive values.

Okay, great. So, now, we have the information we need because we know what ๐ถ๐ธ โ well, more importantly, what ๐ถ๐ธ squared โ is equal to. So, now, letโs move on and find out ๐ต๐ธ. 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 weโve got the triangle ๐ด๐ต๐ธ. So, therefore, what we can say is that ๐ต๐ธ squared is gonna be equal to eight squared plus four squared. So, therefore, ๐ต๐ธ squared is gonna be equal to 64 plus 16. So, therefore, ๐ต๐ธ squared is gonna be equal to 80. So, as before, this is all we need for this problem. However, we couldโve done the same as the last time and taken the root of both sides to find out what ๐ต๐ธ was.

So, now, what we can do is use all the information weโve got to work out whether triangle ๐ต๐ธ๐ถ is, in fact, a right triangle. Because if it is, then ๐ถ๐ธ squared plus ๐ต๐ธ squared is gonna be equal to ๐ต๐ถ squared because ๐ต๐ถ is the longest side. Well, first of all, what weโre gonna do is work out ๐ถ๐ธ squared plus ๐ต๐ธ squared. Well, this is gonna be 20 add 80 cause Iโve shown that we already had ๐ถ๐ธ squared and ๐ต๐ธ squared.

However, if we just have the lengths root 20 and root 80, we couldโve worked this out because root 20 squared would be 20 and root 80 squared would just be 80. And thatโs because weโve got a rule that tells us that if weโve got root ๐ multiplied by root ๐, itโs just equal to ๐. So, root ๐ all squared is just equal to ๐. So, therefore, we know that ๐ถ๐ธ squared plus ๐ต๐ธ 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 ๐ต๐ถ squared is going to be because if this is the same as ๐ถ๐ธ squared plus ๐ต๐ธ squared, then weโve got a right triangle. Well, the square of the longest side ๐ต๐ถ is gonna be equal to 10 squared. So, therefore, ๐ต๐ถ 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 weโve done is weโve 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 ๐ต๐ธ๐ถ is right-angled or a right triangle.

So, in this lesson, weโve covered various different examples, showing different skills. So, weโve had a look at how the Pythagorean theorem is formed. Weโve 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. Weโve also looked at it in this question in a different format that involved a rectangle and different triangles.

And next, weโre gonna have a look at some coordinate geometry.

Two lines intersect at the point ๐ด: three, negative one. One line goes through the point ๐ต: five, one, and the other goes through the point ๐ถ: negative two, six. Find the lengths of the line segments ๐ด๐ต, ๐ด๐ถ, and ๐ต๐ถ.

So, what Iโve done first of all to help us understand what is going on is Iโve drawn a sketch of the three points that weโve got given. So, to find the lengths of our three line segments, what weโre 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 ๐ฅ two minus ๐ฅ one all squared plus ๐ฆ two minus ๐ฆ one all squared. So, itโs the square root of the change in our ๐ฅ-coordinate squared plus the change in our ๐ฆ-coordinate squared.

But where does this formula come from? Well, in fact, itโs an adaptation of the Pythagorean theorem. Because if weโve got two points ๐ฅ one, ๐ฆ one and ๐ฅ two, ๐ฆ two, well, the distance between these two points is, in fact, gonna be the hypotenuse of a right triangle. And thatโs because if we have a look here, if we form a right triangle, weโd have the change of ๐ฅ would be the bottom length and the change of ๐ฆ 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, weโd have our ๐ would be our ๐. And then, we could have our ๐ฅ two minus ๐ฅ one. So, our change in ๐ฅ could be our ๐. And our ๐ฆ two minus ๐ฆ 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 itโs come from, letโs find the lengths of the line segments ๐ด๐ต, ๐ด๐ถ, and ๐ต๐ถ.

So, using this, what we can say is that ๐ด๐ต 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 ๐ฅ-coordinate squared plus the change in our ๐ฆ-coordinate squared. Itโs worth noting that it doesnโt matter which way round theyโre going to be because either way would give us the same result because theyโre 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 ๐ด๐ถ, itโs 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, ๐ต๐ถ can also be found using the same method and itโs also gonna be root 74.

So now weโve answered those parts, letโs move on to the next parts of the question.

So, using the Pythagorean theorem, decide is triangle ๐ด๐ต๐ถ 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 ๐ด๐ถ or ๐ต๐ถ, but in fact theyโre the same length. So, therefore, we cannot have a hypotenuse or longest side with this triangle.

So, therefore, we can say that triangle ๐ด๐ต๐ถ 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 theyโre not at right angles to each other because there is no right triangle.

So, as weโre at the end of this lesson, we can look at the key points. First of all, weโve 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 thatโs because it wouldnโt meet the conditions that are set out by the Pythagorean theorem, which has to apply to right triangles only.