https://file.aiquickdraw.com/mathbot/file/1726105112120r5haaim2.jpg

resuelve aplicando las propiedades de las áreas

To solve the problem shown in the image:

The problem states: "If the triangles ABC and BPQ are equilateral, determine the value of \( x \)." It also includes the diagram where \( PQ = 6 \text{ m} \).

### Step-by-Step Solution:

1. **Understand the Configuration:**
   - The triangles \( ABC \) and \( BPQ \) are both equilateral, meaning all their sides are equal.
   - \( AB = BC = CA = x \) for triangle \( ABC \).
   - \( BP = PQ = QB = 6 \text{ m} \) for triangle \( BPQ \).

2. **Area of Equilateral Triangles:**
   - The area of an equilateral triangle is given by the formula:
     \[
     \text{Area} = \frac{\sqrt{3}}{4} \times \text{side}^2
     \]
   - For triangle \( BPQ \), with side \( 6 \) meters:
     \[
     \text{Area}_{BPQ} = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \text{ square meters}
     \]

3. **Equating the Areas (Since Triangles are Equilateral):**
   - Given that the triangle \( ABC \) is also equilateral, its area can be expressed as:
     \[
     \text{Area}_{ABC} = \frac{\sqrt{3}}{4} \times x^2
     \]
   - Since no specific information about the area of \( ABC \) relative to \( BPQ \) is provided, it's possible that these areas are equal:
     \[
     \frac{\sqrt{3}}{4} \times x^2 = 9\sqrt{3}
     \]
   - Simplifying by dividing both sides by \( \sqrt{3} \) and then by \( \frac{1}{4} \):
     \[
     x^2 = 36
     \]
     \[
     x = 6 \text{ meters}
     \]

Thus, \( x \) is equal to 6 meters.

Would you like more details or have any other questions? Here are five related questions to consider:

1. How would the solution change if \( BPQ \) was not equilateral but only isosceles?
2. What is the height of triangle \( BPQ \) using the given side length?
3. How do the properties of equilateral triangles ensure equal angles?
4. What would be the area of \( ABC \) if \( x \) was twice the length of \( PQ \)?
5. How can the side length of an equilateral triangle be derived from its area?

**Tip:** Remember that the properties of equilateral triangles often simplify calculations involving symmetry and equal lengths, making them useful in geometric problems.

Si los triángulos ABC y BPQ son equiláteros, determina el valor de x.

Determine the value of x when triangles ABC and BPQ are equilateral. Learn how to use the formula for the area of an equilateral triangle and properties of symmetry to solve for x. Suitable for students in Grades 9-11.

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