Question

Would each of them be able to be the side lengths of a triangle?

Answer 1

Pythagoras Theorem:
hipotenuse²=leg₁²+leg₂²

First posible triangle:
hypotenuse=13    (13²=169)
leg₁=12                ( 12²=144)
leg₂=5                  (5²=25)

13³=144 + 25


Answer:can be side lengths of a triangle

Second triangle:
 hypotenuse=12.6    (12.6²=158.76)
leg₁=6.7                ( 6.7²=44.89)
leg₂=6.5                  (6.5²=42.25)

leg₁²+leg₂²=44.89+42.25=87.14≠158.76

Answer: cannot be side lenghts of a triangle.

third triangle:
hypotenuse=13    (13²=169)
leg₁=12                ( 12²=144)
leg₂=11                  (11²=121)

leg₁²+leg₂²=144+121=265≠169

Answer: cannot be side lenghts of a triangle.

fourth triangle:
hypotenuse=13   (13²=169)
leg₁=6                ( 6²=36)
leg₂=4                  (4²=16)

leg₁²+leg₁²=36+16=52≠169

Answer: cannot be side lenghts of a triangle.

Answer 2

Use a bit of logic

the longest side of a triangle has to be less than the sum of the legnths of the other two sides (or else they won't connect or you get a striahge line or something)
baseically
longestside<otherside+shortestside


12,13,5
13<12+5 is that true?
13<17
true
it can be a triangle


6.7,6.5,12.6
12.6<6.7+6.5
12.6<13.2
true, can be a triangle

13,12,11
13<12+11
13<23
true
can be a triangle

13,4,6
13<4+6
13<10
false, cannot be a triangle



all of them can be legnths of a triangle except last one