How do you determine whether the given lengths are sides of a right triangle $ 23,\,18,\,14 $ ?

Hint: First we will consider the lengths of the triangle as $ a,b,c $ . Then square all the sides and arrange them in ascending order. Then according to Pythagoras theorem, hypotenuse square is equal to the sum of square of the sides of the triangle

Complete step-by-step answer:
We will start by considering the lengths of the right-angled triangle as $ a,b,c $ . So, the values of the sides of the triangle will be,
 $
  a = 23 \\
  b = 18 \\
  c = 14 \;
  $
Now, we will arrange these sides according to the ascending order.
 $
  c = 14 \\
  b = 18 \\
  a = 24 \;
  $
Now, according to the Pythagoras Theorem, in a right-angled triangle the hypotenuse square is equal to the sum of squares of the other two sides. Now we will check square all the sides, and check different combinations and check if the sum of any two sides equals the square of the other side.
 $
  {c^2} = 196 \\
  {b^2} = 324 \\
  {a^2} = 576 \;
  $
Now we will make different combinations and try to apply the Pythagoras theorem.
Let us consider the first combination as,
 $
  {a^2} = {b^2} + {c^2} \\
  {24^2} = {18^2} + {14^2} \\
  576 = 324 + 196 \\
  576 = 520 \;
  $
But $ 576 \ne 520 $ .
Hence, the given triangle is not a right-angled triangle.
So, the correct answer is “The given triangle is not a right-angled triangle.”.

Note: Pythagoras theorem is basically used to evaluate the unknown side and unknown angle of a triangle. By this theorem, we can derive a base, perpendicular and hypotenuse formula.
While substituting the terms, assign the values to variables only. While you square the terms, be extra careful in order to avoid any mistakes. Remember that according to the Pythagoras theorem, $ (hypotenuse) = {(side)^2} + {(other\,side)^2} $ . Make sure that the value of the hypotenuse is greater than that of the other two sides.