Video Transcript
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.
