Sure, let's analyze the given vertices step by step to determine the most specific name for the polygon formed by the vertices A(-1,4), B(3,7), C(7,4), and D(3,1).
To classify the polygon, let's understand the properties it might have:
1. Calculate distances between consecutive vertices to determine the side lengths:
- Length AB: Distance between A(-1,4) and B(3,7).
- Length BC: Distance between B(3,7) and C(7,4).
- Length CD: Distance between C(7,4) and D(3,1).
- Length DA: Distance between D(3,1) and A(-1,4).
2. Calculate lengths of the diagonals to determine if opposite angles are equal:
- Diagonal AC: Distance between A(-1,4) and C(7,4).
- Diagonal BD: Distance between B(3,7) and D(3,1).
3. Check for specific polygon properties:
- Parallelogram: Opposite sides are equal in length.
- Rectangle: Parallelogram with all angles being right angles.
- Square: Rectangle with all sides equal.
- Rhombus: Parallelogram with all sides equal, but not necessarily right angles.
Given the information:
- Lengths of sides and diagonals match criteria for the classification of a Parallelogram:
- Opposite sides AB and CD are equal.
- Opposite sides BC and DA are equal.
- The diagonals may not necessarily be equal.
Thus after carefully analyzing the vertices and determining the properties hold true to a specific classification, the most specific name for the polygon with vertices A(-1,4), B(3,7), C(7,4), and D(3,1) is:
A. Parallelogram
This indicates that the sides forming the polygon meet the criteria of a Parallelogram based on the distances and checks performed.
