Let’s find the length and width of the rectangle.

Given:

• Diagonal ($d$) = 5.8 units
• Ratio of length to width ($\frac{\text{length}}{\text{width}}$) = 1 : 2.17

Step-by-Step Solution

1. Define Variables Based on the Ratio: Let the width be $x$ and the length be $2.17x$.

2. Apply the Pythagorean Theorem: For a rectangle with a diagonal $d$, length $L$, and width $W$, $d^2 = L^2 + W^2$ Substituting $L = 2.17x$ and $W = x$: $5.8^2 = (2.17x)^2 + x^2$

3. Solve the Equation:

   • First, calculate $5.8^2$: $5.8^2 = 33.64$
   • Substitute and expand: $33.64 = (2.17x)^2 + x^2$ $33.64 = 4.7089x^2 + x^2$ $33.64 = 5.7089x^2$
   • Solve for $x^2$: $x^2 = \frac{33.64}{5.7089} \approx 5.89$
   • Taking the square root of both sides: $x \approx \sqrt{5.89} \approx 2.43$

4. Calculate the Length and Width:

   • Width $W = x \approx 2.43$
   • Length $L = 2.17x \approx 2.17 \times 2.43 \approx 5.27$

Final Answer:

• Width ≈ 2.43 units
• Length ≈ 5.27 units

Would you like a deeper breakdown of any part? Here are five related questions:

1. How would the calculations change if the diagonal were a different length?
2. What would happen if the length-to-width ratio changed?
3. How is the Pythagorean theorem useful for finding dimensions in a rectangle?
4. Can we use this method for other polygons? Why or why not?
5. What role does squaring and square-rooting play in solving for dimensions?

Tip: For any right triangle problem involving ratios and diagonals, remember that the Pythagorean theorem is essential for linking dimensions.