https://file.aiquickdraw.com/mathbot/file/1726518139119wrb7u7bh.jpg

The problem shown in the image asks for the following:

1. **Calculating the area (luas sawah)** of a rectangular piece of land with dimensions \( (6 + \sqrt{5}) \, \text{m} \) and \( (6 - \sqrt{5}) \, \text{m} \).
2. **Finding the length of the diagonal (panjang diagonal)** of the same rectangle.

Let's solve both parts:

### a) **Calculating the area (Luas sawah)**:
The area \( A \) of a rectangle is given by the formula:
\[
A = \text{length} \times \text{width}
\]
Substituting the given dimensions:
\[
A = (6 + \sqrt{5}) \times (6 - \sqrt{5})
\]
This is a difference of squares, so:
\[
A = 6^2 - (\sqrt{5})^2 = 36 - 5 = 31 \, \text{square meters}
\]

### b) **Finding the diagonal length (Panjang diagonal)**:
The diagonal \( d \) of a rectangle can be found using the Pythagorean theorem:
\[
d = \sqrt{(\text{length})^2 + (\text{width})^2}
\]
Here, the length and width are \( (6 + \sqrt{5}) \) and \( (6 - \sqrt{5}) \). First, calculate the squares of each:

\[
(6 + \sqrt{5})^2 = 36 + 2 \times 6 \times \sqrt{5} + (\sqrt{5})^2 = 36 + 12\sqrt{5} + 5 = 41 + 12\sqrt{5}
\]

\[
(6 - \sqrt{5})^2 = 36 - 2 \times 6 \times \sqrt{5} + (\sqrt{5})^2 = 36 - 12\sqrt{5} + 5 = 41 - 12\sqrt{5}
\]

Now, adding the two expressions:
\[
(6 + \sqrt{5})^2 + (6 - \sqrt{5})^2 = (41 + 12\sqrt{5}) + (41 - 12\sqrt{5}) = 82
\]
Therefore, the diagonal length is:
\[
d = \sqrt{82} \approx 9.06 \, \text{meters}
\]

### Final Answers:
- a) **Area** = 31 square meters
- b) **Diagonal length** ≈ 9.06 meters

Would you like more details or have any questions?

### Related Questions:
1. How is the formula for the area of a rectangle derived?
2. What is the difference of squares, and how is it applied in this problem?
3. How is the Pythagorean theorem used to find the diagonal of a rectangle?
4. Can the area of a rectangle be negative?
5. How does the presence of square roots affect algebraic calculations?

### Tip:
When using the Pythagorean theorem, always square the sides first before summing them up to avoid errors.

Andi menggambar sebidang tanah milik Ayahnya dengan ukuran 6 + √5 meter dan 6 - √5 meter. Hitunglah luas sawah dan panjang diagonal tanah tersebut.

Learn how to calculate the area and diagonal length of a rectangle with dimensions involving square roots. This problem uses the difference of squares and the Pythagorean theorem to find the area and diagonal, with a step-by-step solution. Suitable for students in Grades 8-10.

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