Final answer:
To determine the type of quadrilateral, calculate the slopes of each side. Here, no two sides have equal slopes, indicating the quadrilateral is neither a parallelogram, rectangle, rhombus, nor square. Further analysis is needed for exact classification.
Step-by-step explanation:
To determine the type of quadrilateral formed by the vertices at (2,5), (4,8), (5,3), and (7,6), you can calculate the slopes of the sides. If two opposing sides have equal slopes, then they are parallel and the quadrilateral might be a parallelogram, rectangle, rhombus, or square.
Step-by-step Guide
- Calculate the slope of the side joining (2,5) and (4,8). The slope formula is (y2 - y1) / (x2 - x1). So, slope = (8 - 5) / (4 - 2) = 3 / 2.
- Calculate the slope of the side joining (4,8) and (5,3). Slope = (3 - 8) / (5 - 4) = -5.
- Calculate the slope of the side joining (5,3) and (7,6). Slope = (6 - 3) / (7 - 5) = 3 / 2.
- Calculate the slope of the side joining (7,6) and (2,5). Slope = (5 - 6) / (2 - 7) = -1 / 5.
- Compare slopes for parallelism. If no two sides are parallel, then the quadrilateral could be a trapezoid or a kite, depending on the specific angles and side lengths.
With the calculations, no two sides are parallel; therefore, the quadrilateral does not fall into a parallelogram, rectangle, rhombus, or square category. Other properties such as side lengths and angles would need to be analyzed to label the quadrilateral precisely.