Question

for each set of three lengths, determine if they can be the side lengths of a triangle.
lengths | can be side lengths of a triangle | cannot be side lengths of a triangle
13, 2, 7 | ○ | ○
6, 21, 18 | ○ | ○
5, 3, 19 | ○ | ○
12.5, 5.6, 7.7 | ○ | ○

Explanation:

Step1: Recall triangle inequality rule

For lengths $a,b,c$ (sorted so $c\geq a,c\geq b$), check $a+b > c$.

Step2: Test 13, 2, 7

Sort: 2,7,13. $2+7=9$, $9 < 13$. Fails rule.

Step3: Test 6, 21, 18

Sort: 6,18,21. $6+18=24$, $24 > 21$. Passes rule.

Step4: Test 5, 3, 19

Sort: 3,5,19. $3+5=8$, $8 < 19$. Fails rule.

Step5: Test 12.5, 5.6, 7.7

Sort: 5.6,7.7,12.5. $5.6+7.7=13.3$, $13.3 > 12.5$. Passes rule.

Answer:

Lengths	Can be side lengths of a triangle	Cannot be side lengths of a triangle
6, 21, 18	$\bigcirc$
5, 3, 19		$\bigcirc$
12.5, 5.6, 7.7	$\bigcirc$