9514 1404 393
Answer:
acute
Step-by-step explanation:
I like to think of this in terms of the "form factor" (a term I made up). It is computed as ...
f = a² +b² -c²
where a, b, c are the side lengths, and c is the longest.
The interpretation is ...
f > 0 . . . acute
f = 0 . . . right
f < 0 . . . obtuse
Here, we have ...
f = 13² +17² -21² = 17 . . . . . the triangle is acute
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Additional comment
The measure of the largest angle is ...
arccos(f/(2ab)) = (17/(2(13·17)) = arccos(1/26) ≈ 87.8°
You can see that (2ab) is always positive, so the sign of f matches the sign of the cosine of the angle. That is what leads to the above interpretation.
This formula comes from the Law of Cosines, of which the Pythagorean theorem is a special case.
