Gardening: A gardener wants to split a rectangular bed of flowers diagonally to make two separate triangular beds. What is the area of one of the right triangles if the diagonal is 11 feet and one side is 12 feet?
In a right triangle, the area is given by:

To find the area of one of the right triangles formed by splitting the rectangular bed diagonally, we can follow these steps:

### Step 1: Identify the triangle
The gardener splits the rectangular bed diagonally, forming two right triangles. The sides of the rectangle become the legs of the right triangle, and the diagonal is the hypotenuse.

### Step 2: Use the Pythagorean theorem
The Pythagorean theorem is given by:
\[
c^2 = a^2 + b^2
\]
where \( c \) is the hypotenuse (diagonal), and \( a \) and \( b \) are the legs (sides of the rectangle).

Here:
- \( c = 11 \) feet (the diagonal)
- \( a = 12 \) feet (one side)

We need to find the other side \( b \).

Rearrange the Pythagorean theorem to solve for \( b \):
\[
b = \sqrt{c^2 - a^2} = \sqrt{11^2 - 12^2}
\]

Calculate:
\[
b = \sqrt{121 - 144} = \sqrt{-23}
\]

This gives us a negative value under the square root, indicating that the dimensions given (12 feet and 11 feet diagonal) don't form a valid right triangle. Hence, there is likely a mistake in the problem statement or given dimensions.

### Step 3: Revisiting the Calculation
Let's try recalculating assuming the correct values or reconsider the problem as is:
- \( a = 12 \) feet
- Hypotenuse \( c = 11 \) feet

Since it is impossible to have \( a = 12 \) feet with a smaller diagonal \( c = 11 \) feet, let's check if the problem is meant to involve a triangle with a different side \( b \), or recalculate with different plausible inputs.

### Correct Interpretation (if mistake):
If the hypotenuse \( c \) is greater than either side, for example, \( b = 9 \) feet (adjusted), we can recompute:
\[
c^2 = 12^2 + 9^2 = 144 + 81 = 225 \quad \text{thus, } c = \sqrt{225} = 15 \text{ ft}
\]

Then the area \( A \) for each triangle can be calculated using:
\[
\text{Area } = \frac{1}{2} \times \text{base} \times \text{height}
\]
\[
= \frac{1}{2} \times 12 \times 9 = 54 \text{ square feet}
\]

Please double-check or clarify side values or the hypotenuse for recalculating precisely.

Do you want details or have any questions? 

Here are some related questions:
1. What happens to the area if the diagonal is larger than either side?
2. Can you recalculate the diagonal with different side values?
3. How would the area change if one side is doubled?
4. What would be the area if the side was 11 and diagonal larger?
5. How does Pythagorean theorem explain validity?

**Tip:** Always ensure the hypotenuse is the longest side when using the Pythagorean theorem.

Learn how to calculate the area of right triangles formed by splitting a rectangular flower bed diagonally using the Pythagorean theorem. This problem covers key geometric concepts, including the Pythagorean theorem and the calculation of triangle area. Suitable for students in Grades 7-9.

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