Which of the following can be the sides of a right angled triangle ?

The correct option is 65 cm, 72 cm and 97 cm.

To determine which set of side lengths can form a right-angled triangle, we use the Pythagorean Theorem. According to the theorem, in a right-angled triangle, the square of the longest side (hypotenuse, $c$) is equal to the sum of the squares of the other two sides ($a$ and $b$):

$a^2+b^2=c^2$

Let's check each of the given options:

Option 1: 20 cm, 21 cm, and 31 cm
Here, $a=20$, $b=21$, and $c=31$.
$a^2+b^2={20}^2+{21}^2=400+441=841$
$c^2={31}^2=961$
Since $841\ne 961$, these sides do not form a right-angled triangle.

Option 2: 35 cm, 77 cm, and 88 cm
Here, $a=35$, $b=77$, and $c=88$.
$a^2+b^2={35}^2+{77}^2=1225+5929=7154$
$c^2={88}^2=7744$
Since $7154\ne 7744$, these sides do not form a right-angled triangle.

Option 3: 15 cm, 32 cm, and 57 cm
Here, $a=15$, $b=32$, and $c=57$.
$a^2+b^2={15}^2+{32}^2=225+1024=1249$
$c^2={57}^2=3249$
Since $1249\ne 3249$, these sides do not form a right-angled triangle.

Option 4: 65 cm, 72 cm, and 97 cm
Here, $a=65$, $b=72$, and $c=97$.
$a^2+b^2={65}^2+{72}^2=4225+5184=9409$
$c^2={97}^2=9409$
Since $a^2+b^2=c^2$ ($9409=9409$), the sides 65 cm, 72 cm, and 97 cm satisfy the Pythagorean theorem and form a right-angled triangle.