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: Calculate the sum of squares of two shorter sides and square of the longest side

For a triangle with side lengths \(a = 6\), \(b=10\), \(c = 12\) (where \(c\) is the longest side).
Calculate \(a^{2}+b^{2}\): \(6^{2}+10^{2}=36 + 100=136\)
Calculate \(c^{2}\): \(12^{2}=144\)

Step2: Use the Pythagorean - related inequality for triangle classification

If \(a^{2}+b^{2}=c^{2}\), the triangle is right - angled. If \(a^{2}+b^{2}>c^{2}\), the triangle is acute. If \(a^{2}+b^{2}Since \(136<144\) (i.e., \(6^{2}+10^{2}<12^{2}\)), the triangle is obtuse.
The condition \(a + b>c\) (\(6 + 10>12\)) is just the triangle - inequality (to check if a triangle can exist, not for angle classification).

Answer:

obtuse, because \(6^{2}+10^{2}<12^{2}\)