In a right triangle, the longest side is the hypotenuse. Sum of the squares of the lengths of two other sides is equal to the square of the length of the hypotenuse.

Use right triangle property to determine the lengths which form a right triangle.

(1) 5,11, 13.

The longest side is 13. So,

\(\displaystyle{13}^{{2}}={5}^{{2}}+{11}^{{2}}\)

\(169=25+121\)

\(\displaystyle{169}\ne{146}\)

So, these sides does not form a right triangle.

(2) 9,12,14

The longest side is 14. So,

\(\displaystyle{14}^{{2}}={9}^{{2}}+{12}^{{2}}\)

\(196=81+144\)

\(\displaystyle{196}\ne{225}\)

So, these sides does not form a right triangle.

(3) 8,15,17.

The longest side is 17. So,

\(\displaystyle{17}^{{2}}={8}^{{2}}+{15}^{{2}}\)

\(289=64+225\)

\(289=289\)

So, these sides form the right triangle.

(4) 10,16,19

The longest side is 19. So,

\(\displaystyle{19}^{{2}}={10}^{{2}}+{16}^{{2}}\)

\(361=100+256\)

\(\displaystyle{361}\ne{356}\)

So, these sides does not form a right triangle.