Video: Applying the Converse of Pythagoras’s Theorem

Two lines intersect at the point 𝐴(3, βˆ’1). One line goes through the point 𝐡(5, 1), and the other goes through the point 𝐢(βˆ’2, 6). Find the lengths of the line segments 𝐴𝐡, 𝐴𝐢, and 𝐡𝐢.

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Video Transcript

Two lines intersect at the point 𝐴: three, negative one. One line goes through the point 𝐡: five, one, and the other goes through the point 𝐢: negative two, six. Find the lengths of the line segments 𝐴𝐡, 𝐴𝐢, and 𝐡𝐢.

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 π‘₯ two minus π‘₯ one all squared plus 𝑦 two minus 𝑦 one all squared. So, it’s the square root of the change in our π‘₯-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 π‘₯ one, 𝑦 one and π‘₯ 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 π‘₯ 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 π‘Ž squared plus 𝑏 squared. Well, we’d have our 𝑑 would be our 𝑐. And then, we could have our π‘₯ two minus π‘₯ one. So, our change in π‘₯ could be our π‘Ž. 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 π‘Ž 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 𝐴𝐡, 𝐴𝐢, and 𝐡𝐢.

So, using this, what we can say is that 𝐴𝐡 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 π‘₯-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 π‘Ž, which will simplify to two root two. We did that using a surd relationship.

So then, if we move on to 𝐴𝐢, 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, 𝐡𝐢 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 𝐴𝐡𝐢 a right triangle. And hence are the two lines perpendicular?

As I already stated, the Pythagorean theorem says that 𝑐 squared equals π‘Ž 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 𝐴𝐢 or 𝐡𝐢, 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 𝐴𝐡𝐢 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.

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