Question

a triangle has side lengths of 5,9,and 12 . determine whether the triangle is right , acute or obtuse.

Answer 1

if \(a^2+b^2=c^2\), the triangle is a right triangle, because it satisfies the Pythagorean theorem.

if \(c^2 > a^2+b^2\), the triangle is obtuse
if \(c^2 \lt a^2+b^2\), the triangle is acute

Think of it this way: if you have a triangle where \(c^2 > a^2+b^2\), you have taken a right triangle and pushed the legs surrounding the right angle outward so that you can fit a longer 3rd leg in there. That makes the right angle into an obtuse angle, so you've got an obtuse triangle. Similarly, if you had to push the legs together to connect them to a shorter side, that made the right angle into an acute angle, so you have an acute triangle.