Final answer:
The perimeter of quadrilateral ABCD is approximately 31.5 units, not exactly matching any of the options but closest to 31 units. The midpoint of AD is (3.5, 3.5), and ABCD is not a regular quadrilateral because its sides are not equal. Answer is b. 1. 31 units, 2. (3.5, 3.5), 3. No, because opposite sides are not equal.
Step-by-step explanation:
The student is asking for the perimeter of quadrilateral ABCD, the midpoint of side AD, and whether ABCD is a regular quadrilateral. The perimeter of a polygon is calculated by finding the distance between its consecutive vertices and summing those distances. To calculate the distance between two points, we use the distance formula √((x2-x1)² + (y2-y1)²). The midpoint of a line segment is found by averaging the x-coordinates and the y-coordinates of the endpoints, which gives us the formula ((x1+x2)/2, (y1+y2)/2).
Here, the distances between consecutive vertices are:
- AB = √((10-3)² + (0-0)²) = 7 units
- BC = √((12-10)² + (9-0)²) = √(4 + 81) = √85 ≈ 9.2 units
- CD = √((4-12)² + (7-9)²) = √(64 + 4) = √68 ≈ 8.2 units
- DA = √((3-4)² + (0-7)²) = √(1 + 49) = √50 ≈ 7.1 units
The perimeter of ABCD is the sum of these distances, approximately 7 + 9.2 + 8.2 + 7.1 = 31.5 units. This tells us that the perimeter is closest to option (b).
The midpoint of AD is ((3+4)/2, (0+7)/2) = (3.5, 3.5) which matches options (a) and (b).
A regular quadrilateral, also known as a square, has all sides equal and all angles equal. Since the sides of ABCD are not equal, ABCD is not a regular quadrilateral. The correct answer is (b) with the perimeter being closest to 31 units, midpoint of AD as (3.5, 3.5), and ABCD is not regular because opposite sides are not equal.