### Video Transcript

The triangle π΄π΅πΆ has side
lengths π΄π΅ equals seven centimeters, π΅πΆ equals nine centimeters, and π΄πΆ
equals 10 centimeters. Determine the angle with the
greatest measure in triangle π΄π΅πΆ. And determine the type of
triangle π΄π΅πΆ 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, thatβs the angle at
π΅, 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 π΄πΆ squared is 10 squared,
which is equal to 100. Now the sum of the squares of
the other two sides π΄π΅ and π΅πΆ is seven squared plus nine squared. Thatβs 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 π΅ must be an acute angle. Therefore, we have that π΅ is
the angle with the largest measure. And since π΅ is an acute angle,
triangle π΄π΅πΆ is an acute triangle.