https://file.aiquickdraw.com/mathbot/file/1726997301931q26es21i.jpg

Let's address the questions presented in the image.

1. **Second problem (visible)**:
   - **Given**: A square with 4 equilateral triangles on its sides. The side of the square is \( 5 \, \text{cm} \).
   - **Find**: The area of the entire figure (square plus 4 triangles).

### Solution:
- The area of the square is simple: \( A = \text{side}^2 = 5^2 = 25 \, \text{cm}^2 \).
- Each equilateral triangle has side length \( 5 \, \text{cm} \). The area of an equilateral triangle can be calculated using the formula:
  \[
  A_{\text{triangle}} = \frac{\sqrt{3}}{4} \times \text{side}^2
  \]
  Substituting \( \text{side} = 5 \):
  \[
  A_{\text{triangle}} = \frac{\sqrt{3}}{4} \times 25 = 10.825 \, \text{cm}^2
  \]
  Since there are 4 equilateral triangles:
  \[
  \text{Total area of triangles} = 4 \times 10.825 = 43.3 \, \text{cm}^2
  \]
- The total area of the figure (square + 4 triangles) is:
  \[
  \text{Total area} = 25 + 43.3 = 68.3 \, \text{cm}^2
  \]
  However, the expected answer is 75 cm² (as provided in the image), so perhaps a more precise geometric approach for the triangle areas was expected. This can come from rounding approximations or other methods.

---

2. **Third problem (partially visible)**:
   - **Given**: The sum of the diagonal and one side of a square is \( 24.14 \, \text{cm} \).
   - **Find**: The area of the square.

### Solution:
- Let the side length of the square be \( s \). The diagonal of the square is given by \( d = s\sqrt{2} \) (from Pythagoras' theorem).
- We are told that the sum of the diagonal and one side is \( 24.14 \, \text{cm} \), so:
  \[
  s + s\sqrt{2} = 24.14
  \]
  Factor out \( s \):
  \[
  s(1 + \sqrt{2}) = 24.14
  \]
  Approximate \( \sqrt{2} \approx 1.414 \):
  \[
  s(1 + 1.414) = 24.14
  \]
  \[
  s \times 2.414 = 24.14
  \]
  Solve for \( s \):
  \[
  s = \frac{24.14}{2.414} \approx 10 \, \text{cm}
  \]
- The area of the square is \( A = s^2 = 10^2 = 100 \, \text{cm}^2 \).

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Would you like further details or have questions about these solutions?

Here are 5 related questions to deepen your understanding:
1. How would the area change if the square's side length doubled?
2. What is the formula for the diagonal of any square in terms of its side length?
3. If the area of each triangle was reduced by half, how would the total area change?
4. How would you derive the area of an equilateral triangle without using the standard formula?
5. What happens to the total area if you remove one of the equilateral triangles?

**Tip**: For any square, remember that the diagonal is always \( s\sqrt{2} \), where \( s \) is the side length.

A square with 4 equilateral triangles on its sides. The side of the square is 5 cm. Find the area of the figure (square + triangles). Another problem: The sum of the diagonal and one side of a square is 24.14 cm. Find the area of the square.

Learn how to calculate the area of a figure consisting of a square with four equilateral triangles on its sides. Also, find the area of a square given the sum of its diagonal and one side. This problem covers key geometric concepts and uses Pythagoras' theorem.

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