To find the length of the diagonal walkway across Claire's rectangular garden with dimensions 10.5 feet and 23 feet, we can use the Pythagorean theorem. In a rectangle, the diagonal $d$ forms a right triangle with the two sides $a$ and $b$, where $a = 10.5$ feet and $b = 23$ feet.

The Pythagorean theorem states: $d = \sqrt{a^2 + b^2}$

Step-by-Step Solution

1. Square each side:

   • $a^2 = (10.5)^2 = 110.25$
   • $b^2 = (23)^2 = 529$

2. Add the squares: $a^2 + b^2 = 110.25 + 529 = 639.25$

3. Take the square root of the sum: $d = \sqrt{639.25} \approx 25.3$

Answer

The length of the diagonal is approximately 25.3 feet, rounded to the nearest tenth.

Would you like further details or have any questions?

Here are some related questions to deepen understanding:

1. How would the length of the diagonal change if the garden dimensions were doubled?
2. What would be the diagonal length if the garden dimensions were 15 feet and 20 feet?
3. If the garden was a square with a side length of 23 feet, what would the diagonal be?
4. How does the Pythagorean theorem apply to real-life situations like building walkways or designing rooms?
5. What other ways can you find the diagonal of a rectangle without using the Pythagorean theorem?

Tip: For any rectangle, the diagonal can be quickly calculated by taking the square root of the sum of the squares of its two sides.