To determine the length of the diagonal support for a rectangular metal plate, we can use the Pythagorean theorem. This theorem is applicable in a right-angled triangle where the diagonal of the rectangle forms the hypotenuse, and the sides of the rectangle are the other two sides of the triangle.

The sides of the rectangle are given as 9 cm and 12 cm. According to the Pythagorean theorem, the formula to find the length of the diagonal (hypotenuse) is:

\(c = \sqrt{a^2 + b^2}\)

Here, \(a = 9\) cm and \(b = 12\) cm. Let's substitute these values into the formula:

\(c = \sqrt{(9)^2 + (12)^2}\)

\(c = \sqrt{81 + 144}\)

\(c = \sqrt{225}\)

\(c = 15\)

Thus, the length of the diagonal support needed is 15 cm.

Now, let's verify the given options:

• 16 cm: Incorrect, as it is greater than the calculated value.
• 14 cm: Incorrect, as it is less than the calculated value.
• 13.5 cm: Incorrect, as it is less than the calculated value.
• 15 cm: Correct, as it matches the calculated value.

Therefore, the correct answer is 15 cm.