To determine the type of quadrilateral formed by the vertices [tex]\((2,4)\)[/tex], [tex]\((-4,-2)\)[/tex], [tex]\((-2,4)\)[/tex], and [tex]\((4,-2)\)[/tex], follow these steps:
### Step 1: Calculate the distances between each pair of consecutive vertices
Let's denote the vertices as follows for clarity:
- [tex]\(A = (2, 4)\)[/tex]
- [tex]\(B = (-4, -2)\)[/tex]
- [tex]\(C = (-2, 4)\)[/tex]
- [tex]\(D = (4, -2)\)[/tex]
#### Distance [tex]\(AB\)[/tex]:
[tex]\[
AB = \sqrt{(2 - (-4))^2 + (4 - (-2))^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} = 8.48528137423857
\][/tex]
#### Distance [tex]\(BC\)[/tex]:
[tex]\[
BC = \sqrt{(-4 - (-2))^2 + (-2 - 4)^2} = \sqrt{(-2)^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} = 6.324555320336759
\][/tex]
#### Distance [tex]\(CD\)[/tex]:
[tex]\[
CD = \sqrt{(-2 - 4)^2 + (4 - (-2))^2} = \sqrt{(-6)^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} = 8.48528137423857
\][/tex]
#### Distance [tex]\(DA\)[/tex]:
[tex]\[
DA = \sqrt{(4 - 2)^2 + (-2 - 4)^2} = \sqrt{(2)^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} = 6.324555320336759
\][/tex]
So, the side lengths are:
[tex]\[
AB = 8.48528137423857, \quad BC = 6.324555320336759, \quad CD = 8.48528137423857, \quad DA = 6.324555320336759
\][/tex]
### Step 2: Calculate the lengths of the diagonals
#### Diagonal [tex]\(AC\)[/tex]:
[tex]\[
AC = \sqrt{(2 - (-2))^2 + (4 - 4)^2} = \sqrt{(4)^2 + 0^2} = \sqrt{16} = 4.0
\][/tex]
#### Diagonal [tex]\(BD\)[/tex]:
[tex]\[
BD = \sqrt{(-4 - 4)^2 + (-2 - (-2))^2} = \sqrt{(-8)^2 + 0^2} = \sqrt{64} = 8.0
\][/tex]
So, the diagonals are:
[tex]\[
AC = 4.0, \quad BD = 8.0
\][/tex]
### Step 3: Determine the type of quadrilateral
- Check if all four sides are equal: [tex]\(AB = CD = 8.48528137423857\)[/tex] and [tex]\(BC = DA = 6.324555320336759\)[/tex]. Hence, all four sides are not equal, which excludes it from being a square.
- Check if opposite sides are equal: [tex]\(AB = CD\)[/tex] and [tex]\(BC = DA\)[/tex]. Both pairs of opposite sides are equal.
- Check diagonals: The diagonals [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] are not equal, hence not all diagonals are equal.
Since the quadrilateral has opposite sides equal and diagonals that are not equal, it fits the definition of a parallelogram.
Thus, the type of quadrilateral is:
[tex]\[
\boxed{\text{parallelogram}}
\][/tex]
