This is a right-angled triangle $\triangle ABC$ with $\angle A = 90^\circ$, $AB$ as one leg, $AC = 7\sqrt{3} \, \text{cm}$, and $BC = 14 \, \text{cm}$ (the hypotenuse).

To find the length of side $AB$, we can apply the Pythagorean theorem, which states that in any right triangle:

$AB^2 + AC^2 = BC^2$

Substitute the known values:

$AB^2 + (7\sqrt{3})^2 = 14^2$

First, calculate each term:

$(7\sqrt{3})^2 = 49 \times 3 = 147$ $14^2 = 196$

Now, substitute these into the equation:

$AB^2 + 147 = 196$

Solve for $AB^2$:

$AB^2 = 196 - 147$ $AB^2 = 49$

Finally, take the square root of both sides:

$AB = \sqrt{49} = 7$

Thus, the length of side $AB$ is $\boxed{7 \, \text{cm}}$.

Would you like further details, or do you have any additional questions?

Here are five related questions to explore:

1. How would you find the angles $\angle B$ and $\angle C$ using trigonometric ratios?
2. How can you verify the result using the sine or cosine of one of the angles?
3. What is the area of this right triangle?
4. How would the Pythagorean theorem change if the triangle were not right-angled?
5. Can you use the tangent of $\angle A$ to find the length of $AB$?

Tip: The Pythagorean theorem is only applicable in right-angled triangles. Always verify the triangle type before using it!