In triangle 𝐴𝐵𝐶, suppose that 𝐴𝐵 squared plus 𝐵𝐶 squared equals 72.25 and 𝐴𝐶 equals 8.5, what kind of angle is angle 𝐵? Option A, right. Option B, obtuse. Option C, acute.
Let’s start this question by drawing a diagram of our triangle 𝐴𝐵𝐶. In the question, we can see the notation 𝐴𝐵 squared plus 𝐵𝐶 squared. And this might remind us of a special type of theorem, the Pythagorean theorem, which says that the square of the hypotenuse is equal to the sum of the squares on the other two sides. The Pythagorean theorem applies to right triangles only.
We can write an equation in the Pythagorean theorem for a triangle of lengths 𝑎, 𝑏 and a hypotenuse 𝑐. And that is, that 𝑐 squared equals 𝑎 squared plus 𝑏 squared. So in our diagram, if we have a right angle at angle 𝐵, then we could say that 𝐴𝐵 squared plus 𝐵𝐶 squared would be equal to the hypotenuse 𝐴𝐶 squared.
So now, we need to work out if 𝐴𝐵 squared plus 𝐵𝐶 squared does actually equal 𝐴𝐶 squared. We’re given in the question that 𝐴𝐵 squared plus 𝐵𝐶 squared equals 72.25. And we’re told that the length 𝐴𝐶 is equal to 8.5. So now, we establish if 72.25 equals 8.5 squared. And since we can evaluate that 8.5 squared equals 72.25, then we can say that 𝐴𝐵 squared plus 𝐵𝐶 squared does equal 𝐴𝐶 squared. This means that the type of angle for angle 𝐵 is a right angle.