Final answer:
To determine whether the given vertices form a rectangle, rhombus, or square, calculate the lengths of the sides and the slopes to confirm if they meet the properties of the desired shape: equal sides and right angles for squares, equal opposite sides and right angles for rectangles, or just equal sides for rhombi.
Step-by-step explanation:
In order to determine whether the vertices P(-5,2), Q(4,5), R(6,−1), and S(-3,−4) form a rectangle, rhombus, or square, we must consider the properties of these shapes. A rectangle has all angles equal to 90 degrees, and opposite sides are equal in length. A rhombus has all sides of equal length, but not necessarily all angles equal to 90 degrees. A square has both the properties of a rectangle and a rhombus, meaning all sides equal in length and all angles equal to 90 degrees.
To assess these shapes, we can calculate the lengths of the sides using the distance formula and then the slopes of the lines to check for right angles (i.e., perpendicular lines have slopes that are negative reciprocals). After calculating the distances and slopes, we would compare them to see if any of the mentioned shapes' properties are satisfied. If all sides are equal and the slopes confirm right angles, the vertices form a square. If only the opposite sides are equal and the right angle condition is met, it is a rectangle. If all sides are equal but no right angles, it's a rhombus. If neither condition is fully met, the shape is none of these.