To determine the type of triangle formed by the points [tex]\((3, 2)\)[/tex], [tex]\((-2, -3)\)[/tex], and [tex]\((2, 3)\)[/tex], we start by calculating the distances between each pair of points, which will form the sides of the triangle.
Let's denote these points as follows:
- [tex]\( A = (3, 2) \)[/tex]
- [tex]\( B = (-2, -3) \)[/tex]
- [tex]\( C = (2, 3) \)[/tex]
First, we compute the distances between each pair of points.
1. Distance between [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
AB = \sqrt{(3 - (-2))^2 + (2 - (-3))^2}
\][/tex]
[tex]\[
AB = \sqrt{(5)^2 + (5)^2}
\][/tex]
[tex]\[
AB = \sqrt{25 + 25}
\][/tex]
[tex]\[
AB = \sqrt{50}
\][/tex]
[tex]\[
AB \approx 7.071
\][/tex]
2. Distance between [tex]\( B \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[
BC = \sqrt{((-2) - 2)^2 + ((-3) - 3)^2}
\][/tex]
[tex]\[
BC = \sqrt{(-4)^2 + (-6)^2}
\][/tex]
[tex]\[
BC = \sqrt{16 + 36}
\][/tex]
[tex]\[
BC = \sqrt{52}
\][/tex]
[tex]\[
BC \approx 7.211
\][/tex]
3. Distance between [tex]\( A \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[
AC = \sqrt{(3 - 2)^2 + (2 - 3)^2}
\][/tex]
[tex]\[
AC = \sqrt{(1)^2 + (-1)^2}
\][/tex]
[tex]\[
AC = \sqrt{1 + 1}
\][/tex]
[tex]\[
AC = \sqrt{2}
\][/tex]
[tex]\[
AC \approx 1.414
\][/tex]
We have the lengths of the sides of the triangle:
- [tex]\( AB \approx 7.071 \)[/tex]
- [tex]\( BC \approx 7.211 \)[/tex]
- [tex]\( AC \approx 1.414 \)[/tex]
Next, we ascertain the type of triangle based on these side lengths:
- Equilateral Triangle: All three sides are equal. Here, [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( AC \)[/tex] have different lengths, so the triangle is not equilateral.
- Isosceles Triangle: At least two sides are equal. Here, none of the sides are equal, so the triangle is not isosceles.
- Right Angle Triangle: One of the angles is [tex]\(90^\circ\)[/tex], which can be verified using the Pythagorean theorem. By checking:
- [tex]\( AB^2 \approx (7.071)^2 = 50 \)[/tex]
- [tex]\( BC^2 \approx (7.211)^2 = 52 \)[/tex]
- [tex]\( AC^2 \approx (1.414)^2 = 2 \)[/tex]
We see that none of the conditions [tex]\(AB^2 + AC^2 = BC^2\)[/tex], [tex]\(AB^2 + BC^2 = AC^2\)[/tex], or [tex]\(BC^2 + AC^2 = AB^2\)[/tex] hold true. Thus, the triangle is not a right angle triangle.
- Scalene Triangle: All three sides are of different lengths, and no angles are [tex]\(90^\circ\)[/tex]. This matches our result.
Therefore, the points form a scalene triangle.
Hence, the correct answer is:
b. scalene triangle.
