To find the third side of a triangle, we can use the concept of the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given information:
Side AB = 26 cm
Side BC = 28 cm
Area of the triangle = 336 cm²

Step 1: Finding the altitude
To find the altitude of the triangle, we can use the formula for the area of a triangle:
Area = (1/2) * base * height

In this case, the base can be either side AB or BC, and the corresponding height would be the altitude of the triangle. Let's choose side AB as the base.

336 = (1/2) * 26 * height
height = (336 * 2) / 26
height ≈ 25.85 cm

So, the altitude of the triangle is approximately 25.85 cm.

Step 2: Finding the third side
We can use the Pythagorean theorem to find the length of the third side of the triangle. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Let's assume the third side of the triangle is AC.

AC² = AB² + BC²
AC² = 26² + 28²
AC² = 676 + 784
AC² = 1460

AC ≈ √1460
AC ≈ 38.16 cm

So, the length of the third side AC is approximately 38.16 cm.

Step 3: Checking the triangle inequality theorem
Now, let's check if the sum of the lengths of the two given sides (26 cm and 28 cm) is greater than the length of the third side (38.16 cm).

26 + 28 = 54 cm

Since 54 cm is greater than 38.16 cm, the triangle inequality theorem is satisfied.

Therefore, the correct answer is option C) 30 cm.