Let's denote the sides of the rectangle by $3x$ and $4x$, where $x$ is a common factor.

The perimeter of the rectangle is given by the formula: $\text{Perimeter} = 2 \times (\text{length} + \text{width})$ Substituting the sides into the formula, we have: $98 = 2 \times (3x + 4x)$ Simplifying: $98 = 2 \times 7x$ $98 = 14x$ $x = \frac{98}{14} = 7$

Now that we have $x = 7$, we can find the actual lengths of the sides: $\text{Length} = 3x = 3 \times 7 = 21 \text{ cm}$ $\text{Width} = 4x = 4 \times 7 = 28 \text{ cm}$

The length of the diagonal $d$ can be found using the Pythagorean theorem: $d = \sqrt{(\text{Length})^2 + (\text{Width})^2}$ $d = \sqrt{21^2 + 28^2}$ $d = \sqrt{441 + 784}$ $d = \sqrt{1225}$ $d = 35 \text{ cm}$

So, the length of each diagonal of the rectangle is $35$ cm.

Would you like to explore more details or have any questions?

Here are 5 related questions:

1. What is the area of the rectangle?
2. If the ratio of sides was different, say 2:3, how would the diagonal change?
3. How would the diagonal length change if the perimeter was increased?
4. Can the diagonal length be calculated using trigonometry?
5. How does the ratio of sides affect the properties of the rectangle?

Tip: Understanding the Pythagorean theorem is crucial when working with right-angled shapes like rectangles and triangles.