Certainly, let's demonstrate that the points [tex]\((-3, -3)\)[/tex], [tex]\( (3, 3) \)[/tex], and [tex]\( (-3\sqrt{3}, 3\sqrt{3}) \)[/tex] form an equilateral triangle.
### Step-by-Step Solution
1. Label the Points:
- Let [tex]\( A = (-3, -3) \)[/tex]
- Let [tex]\( B = (3, 3) \)[/tex]
- Let [tex]\( C = (-3\sqrt{3}, 3\sqrt{3}) \)[/tex]
2. Calculate the Distance Between Points:
To determine if the triangle is equilateral, we need to check if the lengths of all three sides are equal. We use the distance formula for this.
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
3. Calculate Distance [tex]\( AB \)[/tex]:
[tex]\[
d_{AB} = \sqrt{(3 - (-3))^2 + (3 - (-3))^2}
\][/tex]
Simplify the terms inside the square root:
[tex]\[
d_{AB} = \sqrt{(3 + 3)^2 + (3 + 3)^2}
\][/tex]
[tex]\[
d_{AB} = \sqrt{6^2 + 6^2}
\][/tex]
[tex]\[
d_{AB} = \sqrt{36 + 36}
\][/tex]
[tex]\[
d_{AB} = \sqrt{72}
\][/tex]
[tex]\[
d_{AB} = 6\sqrt{2} \approx 8.48528137423857
\][/tex]
4. Calculate Distance [tex]\( BC \)[/tex]:
[tex]\[
d_{BC} = \sqrt{(-3\sqrt{3} - 3)^2 + (3\sqrt{3} - 3)^2}
\][/tex]
Simplify the terms inside the square root:
[tex]\[
d_{BC} = \sqrt{(-3\sqrt{3} - 3)^2 + (3\sqrt{3} - 3)^2}
\][/tex]
Since these terms involve squares, which visualize it more clearly:
[tex]\[
d_{BC} = \sqrt{9(3 + 1) + 9(\sqrt{3} - 1)^2}
\][/tex]
Combining these gives:
[tex]\[
d_{BC} = 8.48528137423857
\][/tex]
5. Calculate Distance [tex]\( CA \)[/tex]:
[tex]\[
d_{CA} = \sqrt{(-3\sqrt{3} - (-3))^2 + (3\sqrt{3} - (-3))^2}
\][/tex]
Simplify the terms inside the square root:
[tex]\[
d_{CA} = \sqrt{(-3\sqrt{3} + 3)^2 + (3\sqrt{3} + 3)^2}
\][/tex]
After simplifying:
[tex]\[
d_{CA} = 8.48528137423857
\][/tex]
6. Verify Equal Distances:
[tex]\[
d_{AB} = d_{BC} = d_{CA} = 8.48528137423857
\][/tex]
Since all three sides are equal in length, the triangle formed by points [tex]\((-3, -3)\)[/tex], [tex]\( (3, 3) \)[/tex], and [tex]\( (-3\sqrt{3}, 3\sqrt{3}) \)[/tex] is indeed an equilateral triangle. Therefore, we have shown that these points form an equilateral [tex]\(\Delta^{\text{e}}\)[/tex].
