# Question Video: Applying the Converse of Pythagorasโs Theorem Mathematics • 8th Grade

Two lines intersect at the point ๐ด(3, โ1). One line goes through the point ๐ต(5, 1), and the other goes through the point ๐ถ(โ2, 6). Find the lengths of the line segments ๐ด๐ต, ๐ด๐ถ, and ๐ต๐ถ.

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### Video Transcript

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.