# Question Video: Finding the Unknown Length in a Right-Angled Triangle Using the Euclidean Theorem Mathematics

In the figure shown, if ππΏ = 40 and ππΏ = 30, what is the length of ππ?

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### Video Transcript

In the figure shown, if ππΏ is equal to 40 and ππΏ is 30, what is the length of ππ?

We note first that πΏ is the projection of π onto ππ. And the triangle πππ is a right triangle at π. We can then recall that the corollary to the Euclidean theorem tells us that ππΏ squared is πΏπ multiplied by ππΏ. Now, since we know the lengths of ππΏ and ππΏ, that is, 40 and 30 centimeters, respectively, this will allow us to find πΏπ. We can then apply the Euclidean theorem to find ππ. 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 πΏπ multiplied by 40. Dividing through by 40 and rearranging, this gives us πΏπ 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 ππ squared is equal to πΏπ multiplied by ππ. Remember, πΏπ is the shortest segment of the hypotenuse, which is 22.5 centimeters, and ππ is the length of the hypotenuse of the triangle πππ. We know that the length of the hypotenuse ππ is the sum of the two segments, that is, 40 plus 22.5. And thatβs equal to 62.5. Now, substituting these values into our Euclidean theorem, thatβs ππ squared is equal to πΏπ multiplied by ππ, we have πΏπ is 22.5 multiplied by ππ, which is 62.5. And this evaluates to 1406.25. And now taking the positive square root on both sides, since ππ is a length and so must always be positive, we have that ππ is 37.5 centimeters.

Itβs worth noting that we could also have found this length using the Pythagorean theorem with the right triangle ππΏπ. This gives us ππ squared is equal to πΏπ squared plus ππΏ squared. That is, ππ squared is 22.5 squared plus 30 squared, which is 1406.25 as before. And so again, taking the positive square root, we have ππ is 37.5. And so the length ππ is 37.5 centimeters.