In triangle 𝐴𝐵𝐶, 𝐴𝐵 squared plus 𝐵𝐶 squared is less than 𝐴𝐶 squared. What type of angle is 𝐵?
To solve this problem, we can actually use the Pythagorean theorem and actually use an adaptation of it to help us decide what type of angle 𝐵 is. Well, we know that 𝑎 squared plus 𝑏 squared equals 𝑐 squared. But this is only the case if we have a right triangle. And that’s a triangle with a right angle. And the right angle is opposite our 𝑐, which is our longest side.
And then we have the relationship which is that if 𝑎 squared plus 𝑏 squared is less than 𝑐 squared, then actually we’re gonna have an obtuse triangle. And that means that the angle opposite our longest side is going to be obtuse.
And then finally, we have 𝑎 squared plus 𝑏 squared is greater than 𝑐 squared. This actually shows us that it’s an acute triangle. So, we’re gonna have an acute angle opposite our 𝑐.
Okay, great! So we now have these three relationships that we can use. So let’s use them to help us decide what type of angle 𝐵 is in our triangle 𝐴𝐵𝐶. To help us visualize the problem, I’ve actually just drawn a triangle. It doesn’t have to bear any resemblance to the triangle that we have. It’s just to understand which side is opposite which angle.
So we can see that angle 𝐵 is opposite the side 𝐴𝐶. And as this is the case, we can now work out what type of angle angle 𝐵 is? Because, therefore, as 𝐴𝐵 squared plus 𝐵𝐶 squared is less than 𝐴𝐶 squared, we can look at our second relationship and we can say that, in fact, it’s going to be an obtuse triangle. And therefore, we can say angle 𝐵 is obtuse.