To prove that the triangle with vertices [tex]\( A(-2, 2) \)[/tex], [tex]\( B(2, 2) \)[/tex], and [tex]\( C(4, 2) \)[/tex] is a scalene triangle, we need to show that all three sides of the triangle have different lengths.
Step-by-step solution:
1. Determine the Length of Side AB:
- Coordinates of point [tex]\( A \)[/tex]: [tex]\( (-2, 2) \)[/tex]
- Coordinates of point [tex]\( B \)[/tex]: [tex]\( (2, 2) \)[/tex]
- Using the distance formula [tex]\( AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]:
[tex]\[
AB = \sqrt{(2 - (-2))^2 + (2 - 2)^2} = \sqrt{(2 + 2)^2 + 0^2} = \sqrt{4^2} = \sqrt{16} = 4.0
\][/tex]
2. Determine the Length of Side BC:
- Coordinates of point [tex]\( B \)[/tex]: [tex]\( (2, 2) \)[/tex]
- Coordinates of point [tex]\( C \)[/tex]: [tex]\( (4, 2) \)[/tex]
- Using the distance formula [tex]\( BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]:
[tex]\[
BC = \sqrt{(4 - 2)^2 + (2 - 2)^2} = \sqrt{2^2 + 0^2} = \sqrt{4} = 2.0
\][/tex]
3. Determine the Length of Side CA:
- Coordinates of point [tex]\( C \)[/tex]: [tex]\( (4, 2) \)[/tex]
- Coordinates of point [tex]\( A \)[/tex]: [tex]\( (-2, 2) \)[/tex]
- Using the distance formula [tex]\( CA = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]:
[tex]\[
CA = \sqrt{(4 - (-2))^2 + (2 - 2)^2} = \sqrt{(4 + 2)^2 + 0^2} = \sqrt{6^2} = \sqrt{36} = 6.0
\][/tex]
4. Comparison of Side Lengths:
- Length of [tex]\( AB = 4.0 \)[/tex]
- Length of [tex]\( BC = 2.0 \)[/tex]
- Length of [tex]\( CA = 6.0 \)[/tex]
- We observe that [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( CA \)[/tex] are all different. That is:
[tex]\[
AB \neq BC \neq CA
\][/tex]
Therefore, since all three sides of the triangle have different lengths, the triangle with vertices [tex]\( A(-2, 2) \)[/tex], [tex]\( B(2, 2) \)[/tex], and [tex]\( C(4, 2) \)[/tex] is indeed a scalene triangle.
