Institution: Bangladesh Development Bank Exam: Bangladesh Development Bank Ltd - Senior Officer - 24.11.2017
Topic: Arithmetic Mathematics (পাটিগণিত), Subject: Mathematics (সাধারণ গণিত)

Best Answer Explained in Below-:

In a right-angled triangle, the Pythagorean theorem is fundamental for determining the lengths of the sides. According to this theorem, for a right triangle with sides of lengths \(a\) and \(b\), and hypotenuse \(c\), the relationship can be expressed as:

\[
a^2 + b^2 = c^2
\]

In this specific problem, we have two given sides, one measuring 13 cm and the other 5 cm. We need to identify which side (the hypotenuse or the other leg) corresponds to this configuration. Since the side measuring 13 cm is longer than the other side (5 cm), we logically assume that 13 cm is the hypotenuse \(c\).

Substituting into the Pythagorean theorem:

\[
5^2 + x^2 = 13^2
\]

Calculating this, we get:

\[
25 + x^2 = 169
\]

Now, isolate \(x^2\):

\[
x^2 = 169 - 25
\]
\[
x^2 = 144
\]

Taking the square root of both sides gives us:

\[
x = 12
\]

Thus, the length of the third side is 12 cm. Hence, the correct option is D: equal to 12 cm. This reasoning aligns with the properties of right triangles and the application of the Pythagorean theorem, confirming the validity of our calculations.