### Video Transcript

In the figure shown, if ππΏ is
equal to 40 and ππΏ is 30, what is the length of ππ?

We note first that πΏ is the
projection of π onto ππ. And the triangle πππ is a right
triangle at π. We can then recall that the
corollary to the Euclidean theorem tells us that ππΏ squared is πΏπ multiplied by
ππΏ. Now, since we know the lengths of
ππΏ and ππΏ, that is, 40 and 30 centimeters, respectively, this will allow us to
find πΏπ. We can then apply the Euclidean
theorem to find ππ. So now, substituting in the given
lengths to the corollary of the Euclidean theorem, which is also known as the
altitude rule, we have 30 squared is πΏπ multiplied by 40. Dividing through by 40 and
rearranging, this gives us πΏπ is equal to 30 squared divided by 40, that is, 900
divided by 40, which is 22.5.

So making a note of this and
clearing some space, we can now use the Euclidean theorem to note that ππ squared
is equal to πΏπ multiplied by ππ. Remember, πΏπ is the shortest
segment of the hypotenuse, which is 22.5 centimeters, and ππ is the length of the
hypotenuse of the triangle πππ. We know that the length of the
hypotenuse ππ is the sum of the two segments, that is, 40 plus 22.5. And thatβs equal to 62.5. Now, substituting these values into
our Euclidean theorem, thatβs ππ squared is equal to πΏπ multiplied by ππ, we
have πΏπ is 22.5 multiplied by ππ, which is 62.5. And this evaluates to 1406.25. And now taking the positive square
root on both sides, since ππ is a length and so must always be positive, we have
that ππ is 37.5 centimeters.

Itβs worth noting that we could
also have found this length using the Pythagorean theorem with the right triangle
ππΏπ. This gives us ππ squared is equal
to πΏπ squared plus ππΏ squared. That is, ππ squared is 22.5
squared plus 30 squared, which is 1406.25 as before. And so again, taking the positive
square root, we have ππ is 37.5. And so the length ππ is 37.5
centimeters.