Certainly! Let's find the coordinates of the fourth vertex [tex]\( D \)[/tex] for each of the given problems.
### Problem 1
Given vertices [tex]\( A(-1, 4) \)[/tex], [tex]\( B(7, 4) \)[/tex], and [tex]\( C(7, -1) \)[/tex]:
To find vertex [tex]\( D \)[/tex]:
- The x-coordinate of [tex]\( D \)[/tex] would be [tex]\( A_x + C_x - B_x = -1 + 7 - 7 = -1 \)[/tex].
- The y-coordinate of [tex]\( D \)[/tex] would be [tex]\( A_y + C_y - B_y = 4 + (-1) - 4 = -1 \)[/tex].
Therefore, the coordinates of vertex [tex]\( D \)[/tex] are [tex]\( (-1, -1) \)[/tex].
### Problem 2
Given vertices [tex]\( A(-5, -1) \)[/tex], [tex]\( B(1, -1) \)[/tex], and [tex]\( C(1, -5) \)[/tex]:
To find vertex [tex]\( D \)[/tex]:
- The x-coordinate of [tex]\( D \)[/tex] would be [tex]\( A_x + C_x - B_x = -5 + 1 - 1 = -5 \)[/tex].
- The y-coordinate of [tex]\( D \)[/tex] would be [tex]\( A_y + C_y - B_y = -1 + (-5) - (-1) = -5 \)[/tex].
Therefore, the coordinates of vertex [tex]\( D \)[/tex] are [tex]\( (-5, -5) \)[/tex].
### Problem 3
Given vertices [tex]\( A(2, 3) \)[/tex], [tex]\( B(7, 3) \)[/tex], and [tex]\( C(7, -2) \)[/tex]:
To find vertex [tex]\( D \)[/tex]:
- The x-coordinate of [tex]\( D \)[/tex] would be [tex]\( A_x + C_x - B_x = 2 + 7 - 7 = 2 \)[/tex].
- The y-coordinate of [tex]\( D \)[/tex] would be [tex]\( A_y + C_y - B_y = 3 + (-2) - 3 = -2 \)[/tex].
Therefore, the coordinates of vertex [tex]\( D \)[/tex] are [tex]\( (2, -2) \)[/tex].
### Problem 4
Given vertices [tex]\( A(2, 5) \)[/tex], [tex]\( B(6, 5) \)[/tex], and [tex]\( C(6, 1) \)[/tex]:
To find vertex [tex]\( D \)[/tex]:
- The x-coordinate of [tex]\( D \)[/tex] would be [tex]\( A_x + C_x - B_x = 2 + 6 - 6 = 2 \)[/tex].
- The y-coordinate of [tex]\( D \)[/tex] would be [tex]\( A_y + C_y - B_y = 5 + 1 - 5 = 1 \)[/tex].
Therefore, the coordinates of vertex [tex]\( D \)[/tex] are [tex]\( (2, 1) \)[/tex].
### Summary
- For the first rectangle: [tex]\( D = (-1, -1) \)[/tex].
- For the second rectangle: [tex]\( D = (-5, -5) \)[/tex].
- For the third square: [tex]\( D = (2, -2) \)[/tex].
- For the fourth square: [tex]\( D = (2, 1) \)[/tex].
These steps ensure that the fourth vertex [tex]\( D \)[/tex] is calculated correctly based on the given vertices of the rectangles and squares.
