francisco designed a triangular garden surrounded by three square gardens. the area of garden c (the largest garden) is 400 sq. ft. and the length of a side of garden b is 12 ft. find the lengths of all sides of the triangular gardens and find the areas of the gardens. draw a picture

To solve this problem, we can use some basic geometric principles and formulas. Let's break it down step by step.

1. Draw a picture:
Start by drawing a diagram of the problem description. We have a triangular garden surrounded by three square gardens labeled A, B, and C.

_______
\ A \
\_________\
| |
| Triangular |
| Garden |
| x |
|_________________|
| B |
|__________|
| C |
| 400 sq. ft.|
|__________|

2. Identify what we already know:
We are given that the area of Garden C is 400 square feet and the length of a side of Garden B is 12 feet.

3. Find the side lengths of the triangular garden (x):
Since the triangular garden is surrounded by three square gardens, one of its sides will be the sum of the side lengths of Gardens A and B. Let's find the length of side x.
The length of side B = 12 feet.
The length of side A = The length of side B = 12 feet (since Garden B and Garden C are squares).

Therefore, the length of side x = side A + side B = 12 feet + 12 feet = 24 feet.

4. Find the areas of the gardens:
We have the area of Garden C (400 square feet), and we can easily calculate the areas of the other gardens using their side lengths.

Area of Garden A = (side A)^2 = 12 feet x 12 feet = 144 square feet.
Area of Garden B = (side B)^2 = 12 feet x 12 feet = 144 square feet.
Area of the triangular Garden = (base x height)/2 = (x x x)/2 = (24 feet x 24 feet)/2 = 288 square feet.

5. Summarize the lengths and areas:
The lengths of the gardens are as follows:
- Side A: 12 feet
- Side B: 12 feet
- Side C (triangular garden): 24 feet

The areas of the gardens are as follows:
- Area of Garden A: 144 square feet
- Area of Garden B: 144 square feet
- Area of Garden C: 400 square feet
- Area of the triangular Garden: 288 square feet.

I hope this explanation helps you understand how to solve this problem! Let me know if you have any further questions.