A triangle has sides with lengths 17, 12, and 9. Is the triangle right, acute, or obtuse? Question 1 options:

A)

Can't be determined

B)

Obtuse

C)

Right

D)

Acute

9514 1404 393

  B) obtuse

Step-by-step explanation:

I like to answer these questions using a "form factor" (f) computed as ...

  f = a² + b² - c² . . . . . . . where a, b, c are side lengths with c the longest

Then ...

  f = 9² +12² -17² = 81 +144 -289 = -64

The negative value indicates an obtuse triangle (largest angle more than 90°).

__

f = 0 indicates a right triangle; f > 0 indicates an acute triangle.

Additional comments

You may recognize that the two short sides have the ratio 9:12 = 3:4. You know a (3, 4, 5) triangle is a right triangle, so this triangle would be a right triangle if the longest side were 15. Since it is 17, you know the largest angle is more than 90°.

This "form factor" is related to the angle computation you would do to find the measure of the largest angle using the law of cosines. That angle is ...

  C = arccos(f/(2ab)) = arccos(-64/216) ≈ 107.2°

Since the factor (2ab) is always positive, the sign of "f" tells you the sign of the cosine of the angle. It is negative for angles greater than 90°.