A rectangle \( A B C D \) is to be drawn on a centimetre grid such that \( A \) has coordinates \( (-4,-2) \) \( B \) has coordinates \( (1,10) \) \( C \) has coordinates \( (19, a) \) \( D \) has coordinates \( (b, c) \) Work out the value of \( a \), the value of \( b \) and the value of \( c \). (4 marks) \( \theta a= \) \( \theta b= \) \( \theta c= \)

To find the values of \( a \), \( b \), and \( c \) for the rectangle \( ABCD \), we know the coordinates of points \( A \) and \( B \). Given: - \( A = (-4, -2) \) - \( B = (1, 10) \) To effectively find points \( C \) and \( D \), we can utilize the properties of the rectangle. Specifically, opposite sides of a rectangle are parallel, and the diagonals bisect each other. 1. **Finding \( a \):** The y-coordinates of points \( A \) and \( B \) differ by a certain amount which should also represent the vertical distance between points \( C \) and \( D \). Since \( C \) and \( D \) will vertically align with \( A \) and \( B \), we have: \[ a = -2 + (10 - (-2)) = -2 + 12 = 10 \] 2. **Finding \( b \):** The x-coordinate \( b \) for point \( D \) should equal the x-coordinate of point \( A \). Since \( A \) is \( (-4, -2) \) and the diagonal \( AC \) should equal the diagonal \( BD \): \[ b = -4 \] 3. **Finding \( c \):** Finally, the y-coordinate \( c \) for point \( D \) must be directly aligned with \( B \) since they are opposite corners of the rectangle. \[ c = 10 \] So, the final coordinates are: - \( a = 10 \) - \( b = -4 \) - \( c = 10 \) Thus, we can summarize: \[ \theta a = 10, \theta b = -4, \theta c = 10. \]