Using the Pythagorean Theorem on non-right triangles

Links and useful resources

Lesson-specific resource links

AoPS Online textbook chapter

Concept summary and connections

acute and obtuse angles and the pythagorean theorem
What happens to the pythagorean theorem if we "grow" the hypotenuse of a right triangle a bit?
What happens to it if we "shrink" the hypotenuse a bit?

Growing the hypotenuse makes what was the right angle into an obtuse angle. It also makes the $a^{2} + b^{2} = c^{2}$ equation into an inequality: $a^{2} + b^{2} < c^{2}$ .
Shrinking the hypotenuse has the opposite effect: it makes what was the right angle into an acute angle. It also makes the $a^{2} + b^{2} = c^{2}$ equation into an inequality: $a^{2} + b^{2} > c^{2}$ .

IF there is one side whose square is greater than the sum of the squares of the other two sides, then that triangle is obtuse and the obtuse angle is opposite the long side. On the other hand, if ALL sides haves squares that are less than the sume of the squares of the other two sides, then the triangle is acute. Of course, if one is equal to the sum of the others then the triangle is a right triangle.

We should prove this (in class!)

Lesson and worked examples

\begin{aligned} ∠ C of △ A B C & is acute iff A B^{2} < A C^{2} + B C^{2} \\ ∠ C of △ A B C & is right iff A B^{2} = A C^{2} + B C^{2} \\ ∠ C of △ A B C & is obtuse iff A B^{2} > A C^{2} + B C^{2} \end{aligned}

Media resources

Youtube search for "acute and obtuse angles and the pythagorean theorem"

Guided practice

Is the triangle with sides 5, 12, 14 acute, right, or obtuse?
How about the triangle with sides 2,5,5?

Homework

#hw (geom-gr10) 10.2 Pythagoras – Not Just For Right Triangles? (p. 270) (gr10) 2025-01-27