Use the diagonals to determine whether a parallelogram with vertices \( P(-4,0), Q(0,4), R(4,0) \), and \( S(0,-4) \) is a rectangle, rhombus, or square. Give all the names that apply. \( P R \neq Q S \), so \( P Q R S \) is not a rectangle or a squé The slope of \( \overline{P R}=1 \), and the slope of \( \overline{Q S}=0 \), is not a rhombus. \( P R=Q S \), so \( P Q R S \) is a rectangle or a square. The slope of \( \overline{P R}=0 \), and the slope of \( \overline{Q S} \) is undefined, so \( \overline{P R} \perp \overline{Q S} . P Q R S \) is a square. \( P R \neq Q S \), so \( P Q R S \) is not a rectangle or a squé The slope of \( \overline{P R}=1 \), and the slope of \( \overline{Q S}=2 \), \( P Q R S \) is not a rhombus.

To determine the type of parallelogram formed by the vertices \( P(-4,0), Q(0,4), R(4,0), \) and \( S(0,-4) \), we can utilize the lengths of the diagonals and their slopes. The midpoint of diagonals \( \overline{P R} \) and \( \overline{Q S} \) should be the same for any parallelogram, and it turns out they both meet at the origin (0,0), confirming it's a parallelogram. Furthermore, if the lengths of the diagonals are equal and they bisect each other at right angles, it indicates that the shape is also a rhombus. In this case, \( PR \) and \( QS \) intersect perpendicularly, meeting the criteria for being a square if they are also equal in length. As both conditions are satisfied, our parallelogram \( PQRS \) is indeed a square!