Question Video: Identifying the Angle With the Greatest Meausure in a Triangle Using the Pythagorean Inequality Mathematics • 8th Grade

Triangle 𝐴𝐵𝐶 has side lengths 𝐴𝐵 = 7 cm, 𝐵𝐶 = 9 cm, and 𝐴𝐶 = 10 cm. Determine the angle with the greatest measure in △𝐴𝐵𝐶. Determine the type of △𝐴𝐵𝐶 in terms of its angles.

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Video Transcript

The triangle 𝐴𝐵𝐶 has side lengths 𝐴𝐵 equals seven centimeters, 𝐵𝐶 equals nine centimeters, and 𝐴𝐶 equals 10 centimeters. Determine the angle with the greatest measure in triangle 𝐴𝐵𝐶. And determine the type of triangle 𝐴𝐵𝐶 in terms of its angles.

To answer the first part, we recall that in a triangle, the angle with the greatest measure is always opposite the longest side. Hence, since the longest side in our triangle is 10 centimeters, the angle opposite this, that’s the angle at 𝐵, must have the largest measure.

Now for the second part of the question, to determine the type of triangle we have in terms of its angles, we can apply the Pythagorean inequality theorem, which tells us three things. First, that if the square of the longest side length in a triangle 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 the squares of the other two sides, then the angle opposite is acute. And third, if the square of the longest side equals the sum of the squares of the other two sides, then the angle opposite is a right angle. This third part is actually the converse of the Pythagorean theorem.

Applying this to our triangle, where the longest side is 10 centimeters, we have 𝐴𝐶 squared is 10 squared, which is equal to 100. Now the sum of the squares of the other two sides 𝐴𝐵 and 𝐵𝐶 is seven squared plus nine squared. That’s 49 plus 81, which is equal to 130. We see then that the square of the longest side, which is 100, is less than the sum of the squares of the other two sides, which is 130. So by the Pythagorean inequality theorem, the angle 𝐵 must be an acute angle. Therefore, we have that 𝐵 is the angle with the largest measure. And since 𝐵 is an acute angle, triangle 𝐴𝐵𝐶 is an acute triangle.