Lesson Video: The Converse of the Pythagorean Theorem Mathematics • Second Year of Preparatory School

Video Transcript

So in this lesson, what we’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 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑐 is the hypotenuse, which is the longest side opposite the right angle. And 𝑎 and 𝑏 are the other two sides. And it doesn’t matter which one is which. And it’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 doesn’t work, so 𝑎 squared plus 𝑏 squared isn’t equal to 𝑐 squared, then we can conclude that it won’t be a right triangle.

Okay, great. So, now, we know what it is and how we’re going to use it. Let’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 𝑐 squared equals 𝑎 squared plus 𝑏 squared is necessarily a right triangle? Let us assume that triangle 𝐴𝐵𝐶 is of side lengths 𝑎, 𝑏, and 𝑐 with 𝑐 squared equals 𝑎 squared plus 𝑏 squared. Let triangle 𝐷𝐵𝐶 be a right triangle of side lengths 𝑎, 𝑏, and 𝑑. Part (1) Using the Pythagorean theorem, what can you say about the relationship between 𝑎, 𝑏, and 𝑑?

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

Well, in this orange side here, it’s 𝑑 is our hypotenuse because it’s opposite the right angle and it’s the longest side. So, therefore, as we’re told that triangle 𝐷𝐵𝐶 is a right triangle, this can give us a relationship. And that is that 𝑑 squared is gonna be equal to 𝑎 squared plus 𝑏 squared, which means we’ve answered the first part of the question.

So, now, let’s move on to the second part of the question.

We know that for triangle 𝐴𝐵𝐶, 𝑐 squared equals 𝑎 squared plus 𝑏 squared. What do you conclude about 𝑑?

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

Okay, so, let’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 that’s because if two triangles have the same-length sides, and they are in fact congruent. That’s because that’s one of the ways that we use to prove congruency. Because it’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, we’ve answered this part of the question.

Let’s move on to the next part.

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

Well, we know that triangle 𝐷𝐵𝐶 and triangle 𝐴𝐵𝐶 both have the same length 𝑎. They both have the same length 𝑏. And we’ve already stated that 𝑑 is equal to 𝑐. So, therefore, first of all, we must say that it’s congruent to triangle 𝐷𝐵𝐶. And that’s because all the sides are the same. So, therefore, if they’re congruent, what we can say is that triangle 𝐴𝐵𝐶 has a right angle at 𝐶.

So, what we’ve done here is we’ve shown one way of demonstrating the Pythagorean theorem. And we’ve also answered the question, does this mean that a triangle where 𝑐 squared equals 𝑎 squared plus 𝑏 squared is necessarily a right triangle? And the answer is yes, it is. And we’ve shown that with each of our steps. So, therefore, our full conclusion about triangle 𝐴𝐵𝐶 is that it’s congruent to triangle 𝐷𝐵𝐶. So, it has a right angle at 𝐶.

So, great, we’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 we’re gonna do in this problem is we’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 that’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 couldn’t join them together. So, we couldn’t form a triangle. And if it was equal to 8.1, then it’ll just form a straight line. Okay, great. We know we can form a triangle, but let’s see if we can form a right or right-angled triangle.

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

In rectangle 𝐴𝐵𝐶𝐷, suppose 𝐴𝐸 equals eight, 𝐷𝐸 equals two, and 𝐷𝐶 equals four. Is triangle 𝐵𝐸𝐶 right angled?

Well, the first thing we’re gonna do in this question is add on the information that we know. Well, firstly, we know that 𝐴𝐸 is equal to eight. Then, we know that 𝐷𝐸 is equal to two. And then, we’re told 𝐷𝐶 is equal to four. Okay, great. That’s all the information we’ve been told. But also we can use this to conclude some more information. Well, first of all, 𝐴𝐵 must also be four. That’s cause it’s a rectangle. So, 𝐴𝐵 must be equal to 𝐷𝐶. And 𝐵𝐶 must be equal to two plus eight because it’s these two values added together cause it’s the same as the lengths of 𝐴𝐸 and 𝐷𝐸 together. So, this is gonna have a length of 10.

So, we’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 it’s a rectangle. So, a rectangle would have a right angle at each of its corners. So, what we’re asked to do in this question is work out whether triangle 𝐵𝐸𝐶 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 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared, where 𝑐 is our hypotenuse.

And what we mean by the converse is that if this isn’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 𝐵𝐸𝐶 is, in fact, right angled, well, what we’re gonna need to do is find out each of the lengths. And as we don’t know the lengths 𝐶𝐸 or 𝐵𝐸, we’re gonna have to find out these first.

Well, what we’re gonna do is we’re gonna start with the side 𝐶𝐸. And we can find this out by forming triangle 𝐶𝐷𝐸. And this is a right triangle cause as we’ve shown there’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 𝐶𝐸. So, therefore, we can say that 𝐶𝐸 squared is gonna be equal to two squared plus four squared. And that’s because 𝐶𝐸 is our hypotenuse because it’s opposite the right angle and the longest side. So, therefore, 𝐶𝐸 squared is gonna be equal to four plus 16, which is gonna be equal to 20.

Now, in fact, we don’t have to actually find out what 𝐶𝐸 is. Because when we’re gonna work out the next part, so we’re gonna look at triangle 𝐵𝐸𝐶. We’re gonna have each of the sides squared. So, we can keep this answer we’ve got here. If we wanted to work out what 𝐶𝐸 was, we just take the square root of both sides of this equation here we’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 we’re looking at a length. So, we’re gonna need positive values.

Okay, great. So, now, we have the information we need because we know what 𝐶𝐸 — well, more importantly, what 𝐶𝐸 squared — is equal to. So, now, let’s move on and find out 𝐵𝐸. 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 we’ve got the triangle 𝐴𝐵𝐸. So, therefore, what we can say is that 𝐵𝐸 squared is gonna be equal to eight squared plus four squared. So, therefore, 𝐵𝐸 squared is gonna be equal to 64 plus 16. So, therefore, 𝐵𝐸 squared is gonna be equal to 80. So, as before, this is all we need for this problem. However, we could’ve done the same as the last time and taken the root of both sides to find out what 𝐵𝐸 was.

So, now, what we can do is use all the information we’ve got to work out whether triangle 𝐵𝐸𝐶 is, in fact, a right triangle. Because if it is, then 𝐶𝐸 squared plus 𝐵𝐸 squared is gonna be equal to 𝐵𝐶 squared because 𝐵𝐶 is the longest side. Well, first of all, what we’re gonna do is work out 𝐶𝐸 squared plus 𝐵𝐸 squared. Well, this is gonna be 20 add 80 cause I’ve shown that we already had 𝐶𝐸 squared and 𝐵𝐸 squared.

However, if we just have the lengths root 20 and root 80, we could’ve worked this out because root 20 squared would be 20 and root 80 squared would just be 80. And that’s because we’ve got a rule that tells us that if we’ve got root 𝑎 multiplied by root 𝑎, it’s just equal to 𝑎. So, root 𝑎 all squared is just equal to 𝑎. So, therefore, we know that 𝐶𝐸 squared plus 𝐵𝐸 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 𝐵𝐶 squared is going to be because if this is the same as 𝐶𝐸 squared plus 𝐵𝐸 squared, then we’ve got a right triangle. Well, the square of the longest side 𝐵𝐶 is gonna be equal to 10 squared. So, therefore, 𝐵𝐶 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 we’ve done is we’ve met the criteria for the Pythagorean theorem because 𝑎 squared plus 𝑏 squared is in fact equal to 𝑐 squared. So, therefore, we can say yes, the triangle 𝐵𝐸𝐶 is right-angled or a right triangle.

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

And next, we’re gonna have a look at some coordinate geometry.

Two lines intersect at the point 𝐴: three, negative one. One line goes through the point 𝐵: five, one, and the other goes through the point 𝐶: negative two, six. Find the lengths of the line segments 𝐴𝐵, 𝐴𝐶, and 𝐵𝐶.

So, what I’ve done first of all to help us understand what is going on is I’ve drawn a sketch of the three points that we’ve got given. So, to find the lengths of our three line segments, what we’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 𝑥 two minus 𝑥 one all squared plus 𝑦 two minus 𝑦 one all squared. So, it’s the square root of the change in our 𝑥-coordinate squared plus the change in our 𝑦-coordinate squared.

But where does this formula come from? Well, in fact, it’s an adaptation of the Pythagorean theorem. Because if we’ve got two points 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, well, the distance between these two points is, in fact, gonna be the hypotenuse of a right triangle. And that’s because if we have a look here, if we form a right triangle, we’d have the change of 𝑥 would be the bottom length and the change of 𝑦 would be our vertical length. So, therefore, our hypotenuse would be our 𝑑. So, in that case, if we thought about the Pythagorean theorem, this states that 𝑐 squared equals 𝑎 squared plus 𝑏 squared. Well, we’d have our 𝑑 would be our 𝑐. And then, we could have our 𝑥 two minus 𝑥 one. So, our change in 𝑥 could be our 𝑎. And our 𝑦 two minus 𝑦 one could be our 𝑏.

So, therefore, we can see that in fact, this would be finding 𝑑, our distance, using the Pythagorean theorem. Because if we wanted to find out what 𝑐 was or 𝑑 was, it would in fact just be the square root of 𝑎 squared plus 𝑏 squared, which is what we had at the top. Brilliant! Okay, now, we know the distance formula and where it’s come from, let’s find the lengths of the line segments 𝐴𝐵, 𝐴𝐶, and 𝐵𝐶.

So, using this, what we can say is that 𝐴𝐵 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 𝑥-coordinate squared plus the change in our 𝑦-coordinate squared. It’s worth noting that it doesn’t matter which way round they’re going to be because either way would give us the same result because they’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 𝑎, which will simplify to two root two. We did that using a surd relationship.

So then, if we move on to 𝐴𝐶, it’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, 𝐵𝐶 can also be found using the same method and it’s also gonna be root 74.

So now we’ve answered those parts, let’s move on to the next parts of the question.

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

As I already stated, the Pythagorean theorem says that 𝑐 squared equals 𝑎 squared plus 𝑏 squared, where 𝑐 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 𝐴𝐶 or 𝐵𝐶, but in fact they’re the same length. So, therefore, we cannot have a hypotenuse or longest side with this triangle.

So, therefore, we can say that triangle 𝐴𝐵𝐶 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 they’re not at right angles to each other because there is no right triangle.

So, as we’re at the end of this lesson, we can look at the key points. First of all, we’ve got 𝑐 squared equals 𝑎 squared plus 𝑏 squared. This is Pythagorean theorem, where 𝑐 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 that’s because it wouldn’t meet the conditions that are set out by the Pythagorean theorem, which has to apply to right triangles only.