# Question Video: Determining Whether a Triangle is Obtuse or Acute or a Right Triangle Using Its Side Lengths Mathematics • 8th Grade

Classify the triangle ๐ธ๐น๐ถ, where side length ๐ต๐น = โ3 cm and side length ๐ด๐ธ = โ6 cm and where ๐ด๐ต๐ถ๐ท is a rectangle.

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### Video Transcript

Classify the triangle ๐ธ๐น๐ถ, where side length ๐ต๐น equals root three centimeters and side length ๐ด๐ธ equals root six centimeters and where ๐ด๐ต๐ถ๐ท is a rectangle.

To determine the type of triangle ๐ธ๐น๐ถ, 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 donโt yet know the lengths of the sides of our triangle ๐ธ๐น๐ถ. But we do know that itโs 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 ๐น๐ด๐ธ, since ๐ด๐น is equal to ๐น๐ต , thatโs root three, we can find the length of side ๐ธ๐น. ๐ธ๐น squared equals root six squared plus root three squared, which is nine. And taking the positive square root on both sides โ positive since weโre looking for the length โ we have side length ๐ธ๐น equal to three centimeters.

Now if we consider side length ๐ธ๐ถ, we know that ๐ท๐ธ equals ๐ธ๐ด, which is root six centimeters, and that ๐ท๐ถ equals two root three, since itโs the same length as side ๐ด๐ต. So we have ๐ถ๐ธ squared equals ๐ธ๐ท squared plus ๐ท๐ถ squared. Thatโs root six squared plus two root three squared, which is 18. And taking the positive square root gives ๐ถ๐ธ equal to three root two centimeters.

Using the same method for side ๐ถ๐น, we have ๐ถ๐น squared equals ๐ถ๐ต squared plus ๐ต๐น squared. ๐ถ๐น squared is therefore 27. And so ๐ถ๐น is equal to three root three. We now have all three side lengths for the triangle ๐ธ๐น๐ถ. Now since three root three is greater than three root two which is greater than three, our longest side is ๐ถ๐น.

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 ๐ถ๐น. Thatโs angle ๐ถ๐ธ๐น. We have that ๐ถ๐น squared equals three root three squared, and thatโs 27. Next, we have the sum of the squares of the other two sides, ๐ธ๐น squared plus ๐ถ๐ธ squared, which equals three squared plus three root two squared. And thatโs nine plus 18, which is also equal to 27. Hence, the square of the longest side in triangle ๐ธ๐น๐ถ 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 ๐ถ๐ธ๐น is a right angle.

Now since angle ๐ถ๐ธ๐น is a right angle and the angles of a triangle sum to 180 degrees, the other two angles, ๐ธ๐ถ๐น and ๐ธ๐น๐ถ, must both be acute angles. Hence, as the angle with the largest measure in triangle ๐ธ๐น๐ถ is a right angle, triangle ๐ธ๐น๐ถ is a right triangle.