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The problem is asking for the area of the square \( ABCD \), given the following information:

- \( BE = 3 \, \text{cm} \)
- \( \angle ABE = 60^\circ \)

The diagram shows a square \( ABCD \) with additional points \( E, F, G, H \) inside the square.

### Approach:

We can use trigonometry to solve for the side length of the square \( ABCD \). From the given information:
- \( BE = 3 \, \text{cm} \)
- \( \angle ABE = 60^\circ \)

Since triangle \( ABE \) is a right triangle with a known angle \( \angle ABE = 60^\circ \), we can apply the properties of a 30-60-90 triangle.

In a 30-60-90 triangle, the ratio of the sides is \( 1 : \sqrt{3} : 2 \), corresponding to the shorter leg, longer leg, and hypotenuse, respectively.

- \( BE \) (3 cm) represents the shorter leg opposite the \( 30^\circ \) angle (since \( \angle ABE = 60^\circ \), the remaining angle in triangle \( ABE \) is \( 30^\circ \)).
- The hypotenuse \( AB \) corresponds to twice the length of \( BE \), which would be:
  \[
  AB = 2 \times 3 = 6 \, \text{cm}
  \]

Thus, the side length of square \( ABCD \) is \( AB = 6 \, \text{cm} \).

### Area of the square \( ABCD \):
The area of the square is calculated as:
\[
\text{Area of } ABCD = AB^2 = 6^2 = 36 \, \text{cm}^2
\]

So, the area of \( ABCD \) is \( 36 \, \text{cm}^2 \).

Would you like more details, or do you have any questions about this solution?

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Here are 5 related questions for further practice:

1. How do you calculate the hypotenuse of a right triangle given one leg and an angle?
2. What are the properties of a 30-60-90 triangle?
3. How do you find the area of a square given the length of its diagonal?
4. If the side of the square was doubled, what would the new area be?
5. How would the solution change if the angle \( \angle ABE \) was \( 45^\circ \) instead of \( 60^\circ \)?

**Tip:** When solving for side lengths in right triangles, remembering standard triangle ratios, like those of the 30-60-90 triangle, can significantly simplify your calculations.

정사각형 ABCD에서 BE = 3 cm, ∠ABE = 60°일 때, ABCD의 넓이는?

Learn how to calculate the area of square ABCD, where BE = 3 cm and ∠ABE = 60°. By applying trigonometry, discover the properties of a 30-60-90 triangle and find the square's side length. Suitable for Grades 8-10.

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