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

In the figure shown, if 𝑋𝐿 = 40 and π‘ŒπΏ = 30, what is the length of π‘Œπ‘?


Video Transcript

In the figure shown, if 𝑋𝐿 is equal to 40 and π‘ŒπΏ is 30, what is the length of π‘Œπ‘?

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

It’s worth noting that we could also have found this length using the Pythagorean theorem with the right triangle π‘ŒπΏπ‘. This gives us π‘Œπ‘ squared is equal to 𝐿𝑍 squared plus π‘ŒπΏ squared. That is, π‘Œπ‘ 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‘Œπ‘ is 37.5. And so the length π‘Œπ‘ is 37.5 centimeters.

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