### Video Transcript

Classify the triangle ๐ธ๐น๐ถ,
where side length ๐ต๐น equals root three centimeters and side length ๐ด๐ธ equals
root six centimeters and where ๐ด๐ต๐ถ๐ท is a rectangle.

To determine the type of
triangle ๐ธ๐น๐ถ, we can use the Pythagorean inequality theorem. This tells us that depending on
whether the square of the longest side is greater than, less than, or equal to
the sum of the squares of the other two sides, the angle opposite the longest
side, and therefore the triangle itself, is either obtuse, acute, or right
angled, respectively.

Now in our case, we donโt yet
know the lengths of the sides of our triangle ๐ธ๐น๐ถ. But we do know that itโs
inscribed in a rectangle and that the corners of a rectangle are right
angles. So we can use the Pythagorean
theorem for right-angled triangles to find our three missing side lengths. If we start with triangle
๐น๐ด๐ธ, since ๐ด๐น is equal to ๐น๐ต , thatโs root three, we can find the length
of side ๐ธ๐น. ๐ธ๐น squared equals root six
squared plus root three squared, which is nine. And taking the positive square
root on both sides โ positive since weโre looking for the length โ we have side
length ๐ธ๐น equal to three centimeters.

Now if we consider side length
๐ธ๐ถ, we know that ๐ท๐ธ equals ๐ธ๐ด, which is root six centimeters, and that
๐ท๐ถ equals two root three, since itโs the same length as side ๐ด๐ต. So we have ๐ถ๐ธ squared equals
๐ธ๐ท squared plus ๐ท๐ถ squared. Thatโs root six squared plus
two root three squared, which is 18. And taking the positive square
root gives ๐ถ๐ธ equal to three root two centimeters.

Using the same method for side
๐ถ๐น, we have ๐ถ๐น squared equals ๐ถ๐ต squared plus ๐ต๐น squared. ๐ถ๐น squared is therefore
27. And so ๐ถ๐น is equal to three
root three. We now have all three side
lengths for the triangle ๐ธ๐น๐ถ. Now since three root three is
greater than three root two which is greater than three, our longest side is
๐ถ๐น.

And recalling that the angle
with the largest measure in a triangle is always opposite the longest side, we
can apply the Pythagorean inequality theorem to determine the type of the angle
opposite side ๐ถ๐น. Thatโs angle ๐ถ๐ธ๐น. We have that ๐ถ๐น squared
equals three root three squared, and thatโs 27. Next, we have the sum of the
squares of the other two sides, ๐ธ๐น squared plus ๐ถ๐ธ squared, which equals
three squared plus three root two squared. And thatโs nine plus 18, which
is also equal to 27. Hence, the square of the
longest side in triangle ๐ธ๐น๐ถ is equal to the sum of the squares of the other
two sides. And so by the third part of the
Pythagorean inequality theorem, angle ๐ถ๐ธ๐น is a right angle.

Now since angle ๐ถ๐ธ๐น is a
right angle and the angles of a triangle sum to 180 degrees, the other two
angles, ๐ธ๐ถ๐น and ๐ธ๐น๐ถ, must both be acute angles. Hence, as the angle with the
largest measure in triangle ๐ธ๐น๐ถ is a right angle, triangle ๐ธ๐น๐ถ is a right
triangle.