5. What are the coordinates of \( Y \) if \( (5,12) \) is \( \frac{1}{3} \) of the way from \( X \) to \( Y \) ?

To find the coordinates of \( Y \), we can set up a relationship using the distance. If point \( (5,12) \) is \( \frac{1}{3} \) of the way from \( X \) to \( Y \), we can denote the coordinates of \( X \) as \( (x_1, y_1) \) and \( Y \) as \( (x_2, y_2) \). The point \( (5,12) \) is given by the formula for finding a point that divides a segment in a given ratio. Using the section formula, we can express the coordinates for \( (5,12) \) based on \( X \) and \( Y \): \[ (5, 12) = \left( \frac{2x_1 + x_2}{3}, \frac{2y_1 + y_2}{3} \right) \] From here, we can assume \( X \) has coordinates \( (x_1, y_1) \) and solve for \( (x_2, y_2) \): \[ 5 = \frac{2x_1 + x_2}{3} \quad \Rightarrow \quad 15 = 2x_1 + x_2 \] \[ 12 = \frac{2y_1 + y_2}{3} \quad \Rightarrow \quad 36 = 2y_1 + y_2 \] Solving these two equations will depend on the coordinates of \( X \). For example, if we have \( (x_1, y_1) = (0,0) \), then substituting yields: 1. From \( 15 = 2(0) + x_2 \) leads to \( x_2 = 15 \). 2. From \( 36 = 2(0) + y_2 \) leads to \( y_2 = 36 \). Thus, in this case, \( Y \) would be \( (15, 36) \). The final answer depends on the chosen coordinates of \( X \), but the relationships will remain consistent.