Understanding the Rectangle
To find the coordinates of the other two vertices of a rectangle, we start by recognizing that the diagonals of a rectangle bisect each other. Given the two opposite vertices are P1(-1, 3) and P3(4, 0), we can compute the midpoint (M) of the diagonal.
Calculating the Midpoint
- The midpoint M can be calculated using the formula:
- M = ((x1 + x2) / 2, (y1 + y2) / 2)
- Substituting the coordinates:
- M = ((-1 + 4) / 2, (3 + 0) / 2)
- M = (3/2, 3/2)
Finding the Other Two Vertices
Let’s denote the other two vertices as P2(x2, y2) and P4(x4, y4). Since M is the midpoint of both diagonals, we can use the property of the midpoint to express the coordinates of P2 and P4.
Using Midpoint Properties
- For P2:
- M = ((-1 + x2) / 2, (3 + y2) / 2)
- For P4:
- M = ((4 + x4) / 2, (0 + y4) / 2)
Equating the x-coordinates and y-coordinates gives us:
- From P2:
- -1 + x2 = 3, thus x2 = 4
- 3 + y2 = 3, thus y2 = 0
- From P4:
- 4 + x4 = 3, thus x4 = -1
- 0 + y4 = 3, thus y4 = 3
Conclusion
The other two vertices of the rectangle are:
- P2(4, 0)
- P4(-1, 3)
Thus, the coordinates of the rectangle are:
- P1(-1, 3), P2(4, 0), P3(4, 0), and P4(-1, 3).
This solution shows how to find the vertices of a rectangle using the properties of midpoints and diagonals.