### Video Transcript

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.