To determine if a given quadrilateral is a square, parallelogram, rhombus, or if a point lies on the interior or exterior of a circle, we need to analyze their properties based on their coordinates. Let's go through each statement one by one:
1. Quadrilateral defined by the points (0, -2), (-1, 1), (-3, -3), (-4, 0):
To determine if this quadrilateral is a square, we need to check if its sides are congruent and if its angles are right angles. Calculating the distances between the points, we find that the side lengths are not all equal, so the quadrilateral is not a square.
2. Quadrilateral defined by the points (5, -1), (6, 3), (1, 1), (2, 5):
To determine if this quadrilateral is a parallelogram, we need to check if opposite sides are parallel. By calculating the slopes of the line segments connecting the points, we find that the opposite sides have equal slopes, indicating that they are parallel. Therefore, the quadrilateral is a parallelogram.
3. Quadrilateral defined by the points (7, 2), (5, 4), (4, 1), (2, 3):
To determine if this quadrilateral is a rhombus, we need to check if all sides are congruent. By calculating the distances between the points, we find that not all sides have equal lengths. Therefore, the quadrilateral is not a rhombus.
4. Point (4, -1) on the interior of the circle centered at (3, 2) passing through (1, -1):
To determine if a point lies on the interior of a circle, we need to check if the distance between the point and the center of the circle is less than the radius of the circle. By calculating the distance between (4, -1) and (3, 2), we find that it is less than the distance between (1, -1) and (3, 2), which is the radius of the circle. Therefore, the point (4, -1) lies on the interior of the circle.
5. Point (2, 4) on the exterior of the circle centered at (1, 2) passing through (3, 3):
To determine if a point lies on the exterior of a circle, we need to check if the distance between the point and the center of the circle is greater than the radius of the circle. By calculating the distance between (2, 4) and (1, 2), we find that it is greater than the distance between (3, 3) and (1, 2), which is the radius of the circle. Therefore, the point (2, 4) lies on the exterior of the circle.
In summary:
1. The given quadrilateral is not a square.
2. The given quadrilateral is a parallelogram.
3. The given quadrilateral is not a rhombus.
4. The point (4, -1) lies on the interior of the given circle.
5. The point (2, 4) lies on the exterior of the given circle.