 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 13 feet and one side is 12 feet?

Given that a gardener splits a rectangular bed of flowers diagonally, forming two right triangles, and the diagonal (hypotenuse) is 13 feet while one side (leg) is 12 feet, we can find the area of one of the right triangles as follows:

### Step 1: Identify the known values
- Hypotenuse \( c = 13 \) feet
- One side \( a = 12 \) feet

### Step 2: Use the Pythagorean theorem to find the other side
The Pythagorean theorem states:
\[
c^2 = a^2 + b^2
\]
where \( c \) is the hypotenuse, and \( a \) and \( b \) are the legs of the triangle.

Substitute the known values:
\[
13^2 = 12^2 + b^2
\]
\[
169 = 144 + b^2
\]
\[
b^2 = 169 - 144
\]
\[
b^2 = 25
\]
\[
b = \sqrt{25} = 5 \text{ feet}
\]
So, the other side \( b = 5 \) feet.

### Step 3: Calculate the area of the right triangle
The area \( A \) of a right triangle is given by:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base and height correspond to the two legs \( a \) and \( b \):

\[
\text{Area} = \frac{1}{2} \times 12 \times 5 = \frac{1}{2} \times 60 = 30 \text{ square feet}
\]

### Conclusion
The area of one of the right triangles is **30 square feet**.

Do you want details or have any questions? 

Here are some related questions:
1. How would the area change if both sides of the rectangle were doubled?
2. What would be the area if the other side was 9 feet?
3. How can the diagonal be used to verify rectangle dimensions?
4. What would the area be if only one side was doubled?
5. Can the Pythagorean theorem be applied to other shapes?

**Tip:** Always check if a right triangle follows the Pythagorean theorem to ensure the sides and hypotenuse are correct.

Learn how to calculate the area of triangular beds formed by splitting a rectangular flower bed diagonally. This problem involves applying the Pythagorean theorem and area calculation formulas. Suitable for students in Grades 7-9.

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