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.