To find the most precise name for quadrilateral [tex]\(ABCD\)[/tex] with vertices [tex]\(A(-5,2)\)[/tex], [tex]\(B(-3,5)\)[/tex], [tex]\(C(4,5)\)[/tex], and [tex]\(D(2,2)\)[/tex], we need to analyze the properties of the quadrilateral based on the lengths of its sides and diagonals. Here are the step-by-step details:
1. Calculate the lengths of the sides:
- [tex]\(AB\)[/tex]: The distance between [tex]\(A\)[/tex] and [tex]\(B\)[/tex] is [tex]\(3.605551275463989\)[/tex].
- [tex]\(BC\)[/tex]: The distance between [tex]\(B\)[/tex] and [tex]\(C\)[/tex] is [tex]\(7.0\)[/tex].
- [tex]\(CD\)[/tex]: The distance between [tex]\(C\)[/tex] and [tex]\(D\)[/tex] is [tex]\(3.605551275463989\)[/tex].
- [tex]\(DA\)[/tex]: The distance between [tex]\(D\)[/tex] and [tex]\(A\)[/tex] is [tex]\(7.0\)[/tex].
2. Calculate the lengths of the diagonals:
- [tex]\(AC\)[/tex]: The distance between [tex]\(A\)[/tex] and [tex]\(C\)[/tex] is [tex]\(9.486832980505138\)[/tex].
- [tex]\(BD\)[/tex]: The distance between [tex]\(B\)[/tex] and [tex]\(D\)[/tex] is [tex]\(5.830951894845301\)[/tex].
3. Analyze the side lengths:
- Sides [tex]\(AB\)[/tex] and [tex]\(CD\)[/tex] are equal, both having a length of [tex]\(3.605551275463989\)[/tex].
- Sides [tex]\(BC\)[/tex] and [tex]\(DA\)[/tex] are equal, both having a length of [tex]\(7.0\)[/tex].
4. Analyze the diagonal lengths:
- Diagonals [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] have different lengths, [tex]\(9.486832980505138\)[/tex] and [tex]\(5.830951894845301\)[/tex] respectively.
5. Conclusion:
- Quadrilateral [tex]\(ABCD\)[/tex] has pairs of opposite sides that are equal: [tex]\(AB = CD\)[/tex] and [tex]\(BC = DA\)[/tex].
- The diagonals of the quadrilateral are not equal.
- This property indicates that the quadrilateral is not a rectangle or a square (which would require diagonals to be equal).
- The equal lengths of opposite sides with unequal diagonals strongly suggest that the quadrilateral is a rhombus or kite.
To classify more precisely between a rhombus and kite, we consider the angles and diagonals, but since the diagonals are not equal, and each pair of opposite sides are equal and parallel, the most precise definition fitting these properties is a parallelogram.
[tex]\[
\boxed{\text{parallelogram}}
\][/tex]
