To solve the problem, let us analyze the given details about the rectangle:

1. Perimeter of the rectangle: $16 \, \text{cm}$, so $2 \cdot (\text{length} + \text{width}) = 16 \implies \text{length} + \text{width} = 8.$

2. Length of the diagonal: $6 \, \text{cm}$. By the Pythagorean theorem: $\text{diagonal}^2 = \text{length}^2 + \text{width}^2.$

Thus, we need to solve for the dimensions $l$ (length) and $w$ (width).

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Step 1: Express one variable in terms of the other

From $\text{length} + \text{width} = 8$: $w = 8 - l.$

Step 2: Use the Pythagorean theorem

Substitute $w = 8 - l$ into $\text{length}^2 + \text{width}^2 = \text{diagonal}^2$: $l^2 + (8 - l)^2 = 6^2.$ Simplify: $l^2 + (64 - 16l + l^2) = 36.$ $2l^2 - 16l + 64 = 36.$ $2l^2 - 16l + 28 = 0.$ Divide by 2: $l^2 - 8l + 14 = 0.$

Step 3: Solve the quadratic equation

Use the quadratic formula: $l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \quad \text{where } a = 1, \, b = -8, \, c = 14.$ $l = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(14)}}{2(1)}.$ $l = \frac{8 \pm \sqrt{64 - 56}}{2}.$ $l = \frac{8 \pm \sqrt{8}}{2}.$ $l = \frac{8 \pm 2\sqrt{2}}{2}.$ $l = 4 \pm \sqrt{2}.$

Thus, the possible dimensions are:

• $l = 4 + \sqrt{2}, \, w = 4 - \sqrt{2}$, or vice versa.

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Step 4: Calculate the area

The area of the rectangle is: $\text{Area} = l \cdot w.$ Substitute $l = 4 + \sqrt{2}$ and $w = 4 - \sqrt{2}$: $\text{Area} = (4 + \sqrt{2})(4 - \sqrt{2}).$ $\text{Area} = 4^2 - (\sqrt{2})^2.$ $\text{Area} = 16 - 2 = 14 \, \text{cm}^2.$

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Final Answer:

The area of the rectangle is: $\boxed{14 \, \text{cm}^2}.$

Would you like to explore the geometry of this rectangle further?

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Related Questions:

1. How does the Pythagorean theorem apply to diagonals in other polygons?
2. Can a rectangle's dimensions be determined uniquely given only its perimeter and diagonal?
3. What happens to the area if the diagonal changes but the perimeter remains constant?
4. How can you verify the result graphically or with a physical model?
5. What are the practical implications of solving such a problem in real-world contexts?

Tip: Always recheck quadratic solutions for geometric feasibility (e.g., positive dimensions).