On right triangles, the sides always add up such that [tex]a^2 + b^2 = c^2[/tex] (a and b being the base sides, and c being the hypotenuse). The hypotenuse is always the longest side, so we can push the longest measure in each group to the side until we add up the others.
1. 3, 4, 5
[tex]3^2 = 9
[/tex]
[tex]4^2 = 16[/tex]
[tex]9 + 16 = 25[/tex]
Now let's go to our hypotenuse
[tex]5^2 = 25[/tex]
In this case, [tex]a^2 + b^2[/tex] did indeed equal [tex]c^2[/tex], so this is a right triangle and would go in the YES list.
2. 5, 12, 13
[tex]5^2 = 25[/tex]
[tex]12^2 = 144[/tex]
[tex]25 + 144 = 169[/tex]
[tex]13^2 = 169[/tex]
Again, this is also a right triangle, so it too would go in the YES list.
3. 6, 8, 10
[tex]6^2 = 36
[/tex]
[tex]8^2 = 64[/tex]
[tex]36 + 64 = 100[/tex]
[tex]10^2 = 100[/tex]
Once again, this is a right triangle and belongs in the YES list.
4. 6, 9, 12
[tex]6^2 = 36[/tex]
[tex]9^2 = 81[/tex]
[tex]36 + 81 = 117[/tex]
[tex]12^2 = 144[/tex]
This is not a right triangle and belongs in the NO list.
5. 8, 13, 26
[tex]8^2 = 64[/tex]
[tex]13^2 = 169[/tex]
[tex]64+169=233[/tex]
[tex]26^2 = 676[/tex]
This is also not a right triangle and belongs in the NO list.
6. 8, 15, 17
[tex]8^2 = 64[/tex]
[tex]15^2 =225[/tex]
[tex]64+225 =289[/tex]
[tex]17^2 = 289[/tex]
This is another right triangle that goes in the YES list.
7. 9, 12 14
[tex]9^2 = 81[/tex]
[tex]12^2 = 144[/tex]
[tex]81+144 = 225[/tex]
[tex]14^2 = 196[/tex]
Not a right triangle, goes to NO list.
8. 5/2, 6, 13/2
[tex] \frac{5}{2}^{2} = \frac{25}{4} [/tex]
[tex]6^2 = 36[/tex]
[tex] \frac{25}{4} + 36 = \frac{169}{4} [/tex]
[tex] \frac{13}{2}^{2} = \frac{169}{4}[/tex]
This final one is also indeed a right triangle, and so goes in the YES list.
Hope that helped =)
