### Video Transcript

Suppose that, in triangle ๐ด๐ต๐ถ, ๐ด๐ต equals 13 centimeters, ๐ต๐ถ equals 36 centimeters, and ๐ด๐ถ equals 35 centimeters. What kind of triangle is this in terms of its angles? It is a right-angled triangle, it is an acute-angled triangle, or it is an obtuse-angled triangle.

So weโve been given the three side lengths of a triangle and asked to determine what type of angles it has. In order to do this, we need to look at the relationship between the squares of these three sides.

From the information in the question, we can see that ๐ต๐ถ is the longest side. If the triangle is a right-angled triangle, then the Pythagorean theorem will hold true for its lengths, which means that the square of the longest side, in this case ๐ต๐ถ, will be equal to the sum of the squares of the two shorter sides, ๐ด๐ต and ๐ด๐ถ.

So the triangle will be right-angled if ๐ต๐ถ squared is equal to ๐ด๐ต squared plus ๐ด๐ถ squared. If instead itโs true that ๐ต๐ถ squared is less than the sum of the squares of the other two sides, then the triangle will be acute-angled.

The final possibility is that ๐ต๐ถ squared will be greater than ๐ด๐ต squared plus ๐ด๐ถ squared. And in this case, this will mean that the triangle is obtuse-angled. So letโs evaluate the squares of the three sides in order to determine what type of triangle we have.

๐ต๐ถ squared is equal to 36 squared, which is 1296. ๐ด๐ต squared plus ๐ด๐ถ squared is 13 squared plus 35 squared, which evaluated is 1394. Now letโs compare these values. 1296 is less than 1394. So ๐ต๐ถ squared is less than ๐ด๐ต squared plus ๐ด๐ถ squared. Therefore, we can conclude that this triangle is an acute-angled triangle.