3) The midpoint of \( A B \) is \( M(4,2) \). If the coordinates of \( B \) are \( (6,-4) \), what are the coordinates of \( A \) ?

To find the coordinates of point \( A \), we can use the midpoint formula. The coordinates of the midpoint \( M(x_m, y_m) \) between two points \( A(x_a, y_a) \) and \( B(x_b, y_b) \) can be expressed as: \[ x_m = \frac{x_a + x_b}{2} \quad \text{and} \quad y_m = \frac{y_a + y_b}{2} \] Given that the mid-point \( M(4,2) \) and the coordinates of point \( B(6,-4) \), we can set up the equations: 1. For the x-coordinate: \[ 4 = \frac{x_a + 6}{2} \] 2. For the y-coordinate: \[ 2 = \frac{y_a - 4}{2} \] Now, solving these equations: 1. For \( x_a \): \[ 4 \times 2 = x_a + 6 \implies 8 = x_a + 6 \implies x_a = 2 \] 2. For \( y_a \): \[ 2 \times 2 = y_a - 4 \implies 4 = y_a - 4 \implies y_a = 8 \] Thus, the coordinates of point \( A \) are \( (2, 8) \).