### Video Transcript

Two lines intersect at the point
π΄: three, negative one. One line goes through the point π΅:
five, one, and the other goes through the point πΆ: negative two, six. Find the lengths of the line
segments π΄π΅, π΄πΆ, and π΅πΆ.

So, what Iβve done first of all to
help us understand what is going on is Iβve drawn a sketch of the three points that
weβve got given. So, to find the lengths of our
three line segments, what weβre 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 π¦ two minus π¦ one all squared. So, itβs the square root of the
change in our π₯-coordinate squared plus the change in our π¦-coordinate
squared.

But where does this formula come
from? Well, in fact, itβs an adaptation
of the Pythagorean theorem. Because if weβve got two points π₯
one, π¦ one and π₯ two, π¦ two, well, the distance between these two points is, in
fact, gonna be the hypotenuse of a right triangle. And thatβs because if we have a
look here, if we form a right triangle, weβd have the change of π₯ would be the
bottom length and the change of π¦ would be our vertical length. So, therefore, our hypotenuse would
be our π. So, in that case, if we thought
about the Pythagorean theorem, this states that π squared equals π squared plus π
squared. Well, weβd have our π would be our
π. And then, we could have our π₯ two
minus π₯ one. So, our change in π₯ could be our
π. And our π¦ two minus π¦ one could
be our π.

So, therefore, we can see that in
fact, this would be finding π, our distance, using the Pythagorean theorem. Because if we wanted to find out
what π was or π was, it would in fact just be the square root of π squared plus
π squared, which is what we had at the top. Brilliant! Okay, now, we know the distance
formula and where itβs come from, letβs find the lengths of the line segments π΄π΅,
π΄πΆ, and π΅πΆ.

So, using this, what we can say is
that π΄π΅ 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 π¦-coordinate squared. Itβs worth noting that it doesnβt
matter which way round theyβre going to be because either way would give us the same
result because theyβre 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,
which will simplify to two root two. We did that using a surd
relationship.

So then, if we move on to π΄πΆ,
itβs 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, π΅πΆ can also be found
using the same method and itβs also gonna be root 74.

So now weβve answered those parts,
letβs move on to the next parts of the question.

So, using the Pythagorean theorem,
decide is triangle π΄π΅πΆ a right triangle. And hence are the two lines
perpendicular?

As I already stated, the
Pythagorean theorem says that π squared equals π squared plus π squared, where π
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 π΄πΆ or π΅πΆ, but in fact theyβre the same length. So, therefore, we cannot have a
hypotenuse or longest side with this triangle.

So, therefore, we can say that
triangle π΄π΅πΆ 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 theyβre not at right angles to each other
because there is no right triangle.