Which of the following measurements (in cm) can be the sides of a right triangle ?

The correct option is 14, 50, 48.

To determine which set of side lengths can form a right triangle, we use the Pythagorean Theorem.
According to the Pythagorean Theorem, for a right triangle with side lengths a, b, and hypotenuse c (the longest side):

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

Let's check the given option with side lengths 14 cm, 48 cm, and 50 cm:
Here, the longest side (hypotenuse) is $c=50$ cm, and the other two sides are $a=14$ cm and $b=48$ cm.

First, calculate the sum of the squares of the two shorter sides:

$a^2+b^2={14}^2+{48}^2$

${14}^2=196$

${48}^2=2304$

$a^2+b^2=196+2304=2500$

Now, calculate the square of the longest side:

$c^2={50}^2=2500$

Since $a^2+b^2=c^2$ (2500 = 2500), the measurements 14, 50, 48 satisfy the Pythagorean Theorem and can form a right triangle.