Question

which classification best represents a triangle with side lengths 6 cm, 10 cm, and 12 cm?

• acute, because (6^2 + 10^2 < 12^2)
• acute, because (6 + 10 > 12)
• obtuse, because (6^2 + 10^2 < 12^2)
• obtuse, because (6 + 10 > 12)

Explanation:

Step1: Recall triangle inequality rule

Check if sides form a triangle: $6 + 10 > 12$, $6 + 12 > 10$, $10 + 12 > 6$. This confirms a valid triangle, but does not classify angle type.

Step2: Use Pythagorean theorem extension

For triangle with sides $a \leq b \leq c$, if $a^2 + b^2 < c^2$, it is obtuse. Calculate each term:
$6^2 = 36$, $10^2 = 100$, $12^2 = 144$
Sum of smaller squares: $36 + 100 = 136$
Compare to largest square: $136 < 144$

Step3: Match to correct classification

Since $6^2 + 10^2 < 12^2$, the triangle is obtuse.

Answer:

obtuse, because $6^2 + 10^2 < 12^2$