The line 𝑙:𝑥+3=0 is the axis of symmetry of the equilateral triangle 𝐴𝐵𝐶 . Knowing that 𝐴=(−7,−2) determine the coordinates of the vertices 𝐵 and 𝐶

To find the coordinates of vertices \( B \) and \( C \) of the equilateral triangle \( ABC \), where the line \( l: x + 3 = 0 \) is the axis of symmetry and vertex \( A = (-7, -2) \), follow these steps:

1. **Identify the axis of symmetry**:
   The line \( x + 3 = 0 \) is a vertical line passing through \( x = -3 \).

2. **Reflect vertex \( A \) across the line of symmetry**:
   The x-coordinate of \( A \) is \( -7 \). To find the reflected point across \( x = -3 \):
   \[
   \text{Distance from } -7 \text{ to } -3 = -3 - (-7) = 4
   \]
   So, the x-coordinate of the reflection of \( A \) across \( x = -3 \) will be:
   \[
   -3 + 4 = 1
   \]
   Thus, the reflected point \( A' \) has coordinates \( (1, -2) \).

3. **Determine coordinates of \( B \) and \( C \)**:
   Since \( A \) and \( A' \) are at the base of the equilateral triangle, the third vertex (either \( B \) or \( C \)) must be equidistant from \( A \) and \( A' \).

4. **Calculate the length of \( AA' \)**:
   The length of \( AA' \) is the distance between \( (-7, -2) \) and \( (1, -2) \):
   \[
   d = \sqrt{(1 - (-7))^2 + (-2 - (-2))^2} = \sqrt{(1 + 7)^2 + 0} = \sqrt{8^2} = 8
   \]

5. **Find the height of the equilateral triangle**:
   In an equilateral triangle with side length \( s \), the height \( h \) is given by:
   \[
   h = \frac{\sqrt{3}}{2} s
   \]
   For \( s = 8 \):
   \[
   h = \frac{\sqrt{3}}{2} \cdot 8 = 4\sqrt{3}
   \]

6. **Determine the coordinates of the third vertex**:
   The third vertex is \( 4\sqrt{3} \) units vertically from the midpoint of \( AA' \) which is at \( (-3, -2) \). There are two possible positions for the third vertex \( B \) or \( C \):
   \[
   \text{B or C} = (-3, -2 + 4\sqrt{3}) \quad \text{and} \quad (-3, -2 - 4\sqrt{3})
   \]

Therefore, the coordinates of vertices \( B \) and \( C \) are:
\[
B = (-3, -2 + 4\sqrt{3})
\]
\[
C = (-3, -2 - 4\sqrt{3})
\]

Would you like more details or have any questions?

Here are 5 questions you can ask:
1. How do you reflect a point across a vertical line?
2. Can you explain how to calculate the height of an equilateral triangle?
3. What are the properties of an equilateral triangle?
4. How do you find the midpoint of a line segment?
5. What is the distance formula between two points?

**Tip**: Always verify your reflection by checking the distances on either side of the line of symmetry.

Discover how to determine the coordinates of vertices B and C of equilateral triangle ABC using the axis of symmetry and reflection techniques. This problem involves geometric concepts like reflection across a line, equilateral triangles, and the distance formula. Suitable for Grades 9-12 students.

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