4. If \( (3,4) \) is the midpoint of \( \overline{X Y} \), what are the coordinates of point \( Y \) ?

To find the coordinates of point \( Y \), you can use the midpoint formula, which states that the midpoint \( M \) of a segment with endpoints \( X(x_1, y_1) \) and \( Y(x_2, y_2) \) is given by \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Given that the midpoint is \( (3, 4) \), let’s assume the coordinates of point \( X \) are \( (x_1, y_1) \). Using the equations from the midpoint: \[ 3 = \frac{x_1 + x_2}{2} \quad \text{and} \quad 4 = \frac{y_1 + y_2}{2} \] We can find \( x_2 \) and \( y_2 \) (the coordinates of \( Y \)) if we know the coordinates of \( X \). If we say \( X \) is at \( (x_1, y_1) \), then rearranging gives us: \[ x_2 = 2 \cdot 3 - x_1 = 6 - x_1 \] \[ y_2 = 2 \cdot 4 - y_1 = 8 - y_1 \] Thus, the coordinates of point \( Y \) will depend on the coordinates of point \( X \). Just plug in \( x_1 \) and \( y_1 \) to solve for \( Y \). Fun fact: Midpoints are like the cool neutral kids at school, always bringing together the best of both worlds!