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

Video Transcript

𝐴𝐵𝐶𝐷 is a parallelogram. If 𝐴𝐶 equals 13 centimeters, 𝐴𝐷 equals 13 centimeters, and 𝐷𝐶 equals five centimeters, what is the type of triangle 𝐴𝐷𝐶?

We see that triangle 𝐴𝐷𝐶 is an isosceles triangle. And recalling that the angle in a triangle with the greatest measure is opposite the longest side, in triangle 𝐴𝐷𝐶 the angles at 𝐶 and 𝐷, which are equal, will have the largest measure. Choosing either one of the angles at 𝐶 and 𝐷, we can use the Pythagorean inequality theorem to confirm that these angles are acute.

Taking the angle at 𝐷 to work on, this theorem tells us three things. First, that if the square of the longest side is greater than the sum of the squares of the other two sides, then the angle opposite the longest side is an obtuse angle. Second, if the square of the longest side is less than the sum of squares of the other two sides, the angle is acute. And third, if the square of the longest side is equal to the sum of the squares of the other two, then the angle opposite is a right angle.

In our case, we have 𝐴𝐶 squared, that is 13 squared, equals 169 and that 𝐴𝐷 squared plus 𝐷𝐶 squared equals 13 squared plus five squared. And that’s equal to 194. Hence, 𝐴𝐶 squared is less than 𝐴𝐷 squared plus 𝐷𝐶 squared. And so angle 𝐶𝐷𝐴 is an acute angle. Angle 𝐴𝐶𝐷 is the same, so this is also acute. And since these angles have the largest measure in triangle 𝐴𝐷𝐶, angle 𝐶𝐴𝐷 must be smaller than them. Hence, the third angle, angle 𝐶𝐴𝐷 is also acute. Since all three angles are acute and, in particular, the angle with the largest measure is acute, triangle 𝐴𝐷𝐶 is an acute triangle.