What is the value of \( r \) if the midpoent bethecen the points \( [4,2] \) and \( [1, r] \) is \( \left[\frac{5}{2}, \frac{9}{2}\right] \)

To find the value of \( r \), we use the formula for the midpoint of two points \((x_1, y_1)\) and \((x_2, y_2)\), which is given by: \[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] In this case, the points are \([4, 2]\) and \([1, r]\). We can identify \(x_1 = 4\), \(y_1 = 2\), \(x_2 = 1\), and \(y_2 = r\). The given midpoint is \(\left[ \frac{5}{2}, \frac{9}{2} \right]\). Using the x-coordinates for the midpoint: \[ \frac{4 + 1}{2} = \frac{5}{2} \] This confirms the x-coordinate is correct. Now, we'll work on the y-coordinates: \[ \frac{2 + r}{2} = \frac{9}{2} \] To eliminate the denominator, we multiply both sides by 2: \[ 2 + r = 9 \] Now, we solve for \(r\): \[ r = 9 - 2 \] \[ r = 7 \] Thus, the value of \( r \) is \( 7 \).