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Let's determine whether the points [tex]\( (-3, 3) \)[/tex], [tex]\( (3, 3) \)[/tex], and [tex]\( (-3\sqrt{3}, 3\sqrt{3}) \)[/tex] form an equilateral triangle by calculating the distances between each pair of points and comparing these distances.
### Step 1: Calculate the distance between the first and second points [tex]\((-3, 3)\)[/tex] and [tex]\((3, 3)\)[/tex].
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For points [tex]\((-3, 3)\)[/tex] and [tex]\((3, 3)\)[/tex]:
[tex]\[ d_{1} = \sqrt{(3 - (-3))^2 + (3 - 3)^2 } = \sqrt{(3 + 3)^2 + 0^2} = \sqrt{6^2} = \sqrt{36} = 6 \][/tex]
### Step 2: Calculate the distance between the second and third points [tex]\((3, 3)\)[/tex] and [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex].
For points [tex]\((3, 3)\)[/tex] and [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex]:
[tex]\[ d_{2} = \sqrt{(-3\sqrt{3} - 3)^2 + (3\sqrt{3} - 3)^2} \][/tex]
Let's calculate the terms individually:
[tex]\[ (-3\sqrt{3} - 3)^2 = ( -3\sqrt{3} - 3)^2 = 9 \cdot 3 + 2 \cdot -3 \cdot 3\sqrt{3} + 3^2 = 27 + 2( -9\sqrt{3}) + 9 = 27 - 18\sqrt{3} + 9 = 36 - 18 \sqrt{3} \][/tex]
[tex]\[ (3\sqrt{3} - 3)^2 = (3\sqrt{3} - 3)^2 = 9 \cdot 3 - 2 \cdot 3 \cdot 3 \cdot \sqrt{3} + 9 \\ = 27 - 18 \sqrt{3}+ 9 = 36 - 18\sqrt{3} \][/tex]
[tex]\[ d_{2} = \sqrt{36 - 18\sqrt{3} + 27 - 36 + (18\sqrt{274)} }} = \sqrt (36 -18\sqrt(3) + 27) = \sqrt(63) = 8.485 \][/tex]
### Step 3: Calculate the distance between the third and first points [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex] and [tex]\((-3, 3)\)[/tex].
For points [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex] and [tex]\((-3,3)\)[/tex]:
[tex]\[ d_{3} = \sqrt{(-3 - (-3\sqrt{3}))^2 + (3 - 3\sqrt{3})^2} \][/tex]
Let's calculate the terms individually:
[tex]\[ (-3 - (-3\sqrt{3}))^2 = (-3 +3\sqrt{3})^2 = 36 - 18 \sqrt{9} = (- 6\sqrt( {3})\][/tex]
[tex]\[ (3 - 3\sqrt{3})^2 = (3 -3\sqrt3) = - 9 \sqrt{3}\][/tex]
[tex]\[ d_{3} = \sqrt 6 - 3 \sqrt{3} and sqrt(4) 36 ==}} = 3.104 \][/tex]
### Step 4: Compare the distances
We have found the following distances:
- Distance between [tex]\((-3, 3)\)[/tex] and [tex]\((3, 3)\)[/tex]: [tex]\( 6 \)[/tex]
- Distance between [tex]\((3, 3)\)[/tex] and [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex]: [tex]\( 8.485\)[/tex]
- Distance between [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex] and [tex]\((-3, 3)\)[/tex]: [tex]\( 3.104\)[/tex]
By comparing these distances, we observe that they are not equal.
Therefore, the triangle formed by the points [tex]\((-3, 3)\)[/tex], [tex]\( (3, 3) \)[/tex], and [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex] is not an equilateral triangle.
### Step 1: Calculate the distance between the first and second points [tex]\((-3, 3)\)[/tex] and [tex]\((3, 3)\)[/tex].
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For points [tex]\((-3, 3)\)[/tex] and [tex]\((3, 3)\)[/tex]:
[tex]\[ d_{1} = \sqrt{(3 - (-3))^2 + (3 - 3)^2 } = \sqrt{(3 + 3)^2 + 0^2} = \sqrt{6^2} = \sqrt{36} = 6 \][/tex]
### Step 2: Calculate the distance between the second and third points [tex]\((3, 3)\)[/tex] and [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex].
For points [tex]\((3, 3)\)[/tex] and [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex]:
[tex]\[ d_{2} = \sqrt{(-3\sqrt{3} - 3)^2 + (3\sqrt{3} - 3)^2} \][/tex]
Let's calculate the terms individually:
[tex]\[ (-3\sqrt{3} - 3)^2 = ( -3\sqrt{3} - 3)^2 = 9 \cdot 3 + 2 \cdot -3 \cdot 3\sqrt{3} + 3^2 = 27 + 2( -9\sqrt{3}) + 9 = 27 - 18\sqrt{3} + 9 = 36 - 18 \sqrt{3} \][/tex]
[tex]\[ (3\sqrt{3} - 3)^2 = (3\sqrt{3} - 3)^2 = 9 \cdot 3 - 2 \cdot 3 \cdot 3 \cdot \sqrt{3} + 9 \\ = 27 - 18 \sqrt{3}+ 9 = 36 - 18\sqrt{3} \][/tex]
[tex]\[ d_{2} = \sqrt{36 - 18\sqrt{3} + 27 - 36 + (18\sqrt{274)} }} = \sqrt (36 -18\sqrt(3) + 27) = \sqrt(63) = 8.485 \][/tex]
### Step 3: Calculate the distance between the third and first points [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex] and [tex]\((-3, 3)\)[/tex].
For points [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex] and [tex]\((-3,3)\)[/tex]:
[tex]\[ d_{3} = \sqrt{(-3 - (-3\sqrt{3}))^2 + (3 - 3\sqrt{3})^2} \][/tex]
Let's calculate the terms individually:
[tex]\[ (-3 - (-3\sqrt{3}))^2 = (-3 +3\sqrt{3})^2 = 36 - 18 \sqrt{9} = (- 6\sqrt( {3})\][/tex]
[tex]\[ (3 - 3\sqrt{3})^2 = (3 -3\sqrt3) = - 9 \sqrt{3}\][/tex]
[tex]\[ d_{3} = \sqrt 6 - 3 \sqrt{3} and sqrt(4) 36 ==}} = 3.104 \][/tex]
### Step 4: Compare the distances
We have found the following distances:
- Distance between [tex]\((-3, 3)\)[/tex] and [tex]\((3, 3)\)[/tex]: [tex]\( 6 \)[/tex]
- Distance between [tex]\((3, 3)\)[/tex] and [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex]: [tex]\( 8.485\)[/tex]
- Distance between [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex] and [tex]\((-3, 3)\)[/tex]: [tex]\( 3.104\)[/tex]
By comparing these distances, we observe that they are not equal.
Therefore, the triangle formed by the points [tex]\((-3, 3)\)[/tex], [tex]\( (3, 3) \)[/tex], and [tex]\((-3\sqrt{3}, 3\sqrt{3})\)[/tex] is not an equilateral triangle.
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