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

Classify the triangle 𝐸𝐹𝐶, where side length 𝐵𝐹 = √3 cm and side length 𝐴𝐸 = √6 cm and where 𝐴𝐵𝐶𝐷 is a rectangle.

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Video Transcript

Classify the triangle 𝐸𝐹𝐶, where side length 𝐵𝐹 equals root three centimeters and side length 𝐴𝐸 equals root six centimeters and where 𝐴𝐵𝐶𝐷 is a rectangle.

To determine the type of triangle 𝐸𝐹𝐶, 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 don’t yet know the lengths of the sides of our triangle 𝐸𝐹𝐶. But we do know that it’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 𝐹𝐴𝐸, since 𝐴𝐹 is equal to 𝐹𝐵 , that’s root three, we can find the length of side 𝐸𝐹. 𝐸𝐹 squared equals root six squared plus root three squared, which is nine. And taking the positive square root on both sides — positive since we’re looking for the length — we have side length 𝐸𝐹 equal to three centimeters.

Now if we consider side length 𝐸𝐶, we know that 𝐷𝐸 equals 𝐸𝐴, which is root six centimeters, and that 𝐷𝐶 equals two root three, since it’s the same length as side 𝐴𝐵. So we have 𝐶𝐸 squared equals 𝐸𝐷 squared plus 𝐷𝐶 squared. That’s root six squared plus two root three squared, which is 18. And taking the positive square root gives 𝐶𝐸 equal to three root two centimeters.

Using the same method for side 𝐶𝐹, we have 𝐶𝐹 squared equals 𝐶𝐵 squared plus 𝐵𝐹 squared. 𝐶𝐹 squared is therefore 27. And so 𝐶𝐹 is equal to three root three. We now have all three side lengths for the triangle 𝐸𝐹𝐶. Now since three root three is greater than three root two which is greater than three, our longest side is 𝐶𝐹.

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 𝐶𝐹. That’s angle 𝐶𝐸𝐹. We have that 𝐶𝐹 squared equals three root three squared, and that’s 27. Next, we have the sum of the squares of the other two sides, 𝐸𝐹 squared plus 𝐶𝐸 squared, which equals three squared plus three root two squared. And that’s nine plus 18, which is also equal to 27. Hence, the square of the longest side in triangle 𝐸𝐹𝐶 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 𝐶𝐸𝐹 is a right angle.

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