Okay, for the L-shape, the smaller rectangle is 32 ft. x 24.31 ft. with a area of 778 sq. ft. The larger rectangle is 55 ft. x 20 ft. with a area of 1,100 sq. ft. The longer side of the L is 55 ft. and the shorter side is 32 ft. How long are the other sides on the shape?

To find the lengths of the other sides of the L-shape, we'll use the dimensions and areas of the two rectangles that make up the shape.

### Given:
- **Smaller Rectangle:** 32 ft × 24.31 ft with an area of 778 sq. ft.
- **Larger Rectangle:** 55 ft × 20 ft with an area of 1,100 sq. ft.
- **Longer side of the L-shape:** 55 ft.
- **Shorter side of the L-shape:** 32 ft.

### Analysis:
1. The longer side of the L-shape corresponds to the longer side of the larger rectangle, which is 55 ft.
2. The shorter side of the L-shape corresponds to the shorter side of the smaller rectangle, which is 32 ft.
3. To determine the lengths of the other two sides of the L-shape:
   - The length of the side parallel to the 24.31 ft side of the smaller rectangle (let’s call this \( L_1 \)).
   - The length of the side parallel to the 20 ft side of the larger rectangle (let’s call this \( L_2 \)).

### Finding \( L_1 \) and \( L_2 \):
- \( L_1 \) (the missing side adjacent to the smaller rectangle): This side will be the difference between the length of the larger rectangle (55 ft) and the shorter side of the smaller rectangle (32 ft).
  
  \( L_1 = 55 \text{ ft} - 32 \text{ ft} = 23 \text{ ft} \)

- \( L_2 \) (the missing side adjacent to the larger rectangle): This side will be the difference between the width of the smaller rectangle (24.31 ft) and the width of the larger rectangle (20 ft).
  
  \( L_2 = 24.31 \text{ ft} - 20 \text{ ft} = 4.31 \text{ ft} \)

### Conclusion:
The lengths of the other sides of the L-shape are:
- One side is \( 23 \) ft.
- The other side is \( 4.31 \) ft.

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How would the areas change if the dimensions of the rectangles were adjusted?
2. What would the perimeter of the L-shape be?
3. How does the area of the L-shape compare to the sum of the areas of the rectangles?
4. How would you calculate the area of the L-shape directly without breaking it into rectangles?
5. What if the longer side of the L-shape were doubled—how would the other sides change?

**Tip:** When working with composite shapes like an L-shape, breaking them down into simpler rectangles can make calculations easier.

Learn how to find the lengths of the other sides of an L-shape formed by two rectangles. Understand the properties of rectangles and apply area calculations to determine the missing dimensions. Suitable for high school students studying geometry and area calculations.

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