How do you determine the lengths 8,31.5,32.5 from a right triangle?
(a) By Pythagoras theorem
(b) By adding the lengths
(c) By subtracting the lengths
(d) None of these

Hint: We have a problem where we have to check if a triangle with given lengths 8,31.5,32.5 form a right angle triangle or not. So, we are going to use the Pythagoras theorem to check that fact. By putting the values of lengths in the formula and analyzing them we will find our desired result.

Complete step by step solution:
We are to check if the lengths 8,31.5,32.5 form a right triangle. We are going to prove that from Pythagoras’s theorem.
In mathematics, the Pythagorean theorem, or Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation.

Here ${{c}^{2}}={{a}^{2}}+{{b}^{2}}$, where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. The theorem, whose history is the subject of much debate, is named for the Greek thinker Pythagoras, born around 570 BC.
So, clearly, $c=32.5$, $a=31.5$ and $b=8$
Now, we are to check if ${{c}^{2}}={{a}^{2}}+{{b}^{2}}$,
${{c}^{2}}={{\left( 32.5 \right)}^{2}}=1056.25$
And,
${{a}^{2}}+{{b}^{2}}={{\left( 8 \right)}^{2}}+{{\left( 31.5 \right)}^{2}}=64+992.25=1056.25$
Hence, we can see, ${{c}^{2}}={{a}^{2}}+{{b}^{2}}=1056.25$,
Thus, the lengths of the triangles form a right angle triangle.

Hence the solution is, (a) By Pythagoras theorem.

Note:
Pythagoras theorem is usually useful to find the sides of a right-angled triangle like 8,31.5,32.5 here. If we know the two sides of a right triangle, then we can find the third side. Also To find the diagonal of a square, Pythagoras theorem is used.