Suppose that, in triangle 𝐴𝐵𝐶, 𝐴𝐵 equals 13 centimeters, 𝐵𝐶 equals 36 centimeters, and 𝐴𝐶 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 we’ve 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 𝐵𝐶 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 𝐵𝐶, will be equal to the sum of the squares of the two shorter sides, 𝐴𝐵 and 𝐴𝐶.
So the triangle will be right-angled if 𝐵𝐶 squared is equal to 𝐴𝐵 squared plus 𝐴𝐶 squared. If instead it’s true that 𝐵𝐶 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 𝐵𝐶 squared will be greater than 𝐴𝐵 squared plus 𝐴𝐶 squared. And in this case, this will mean that the triangle is obtuse-angled. So let’s evaluate the squares of the three sides in order to determine what type of triangle we have.
𝐵𝐶 squared is equal to 36 squared, which is 1296. 𝐴𝐵 squared plus 𝐴𝐶 squared is 13 squared plus 35 squared, which evaluated is 1394. Now let’s compare these values. 1296 is less than 1394. So 𝐵𝐶 squared is less than 𝐴𝐵 squared plus 𝐴𝐶 squared. Therefore, we can conclude that this triangle is an acute-angled triangle.