1. Find the midpoint of the line segment joining (2, 7) and (10, 3):

\(\left( \frac{2 + 10}{2}, \frac{7 + 3}{2} \right) = (6, 5)\)

2. Calculate the gradient (slope) of the line segment:

\(m = \frac{3 - 7}{10 - 2} = -\frac{1}{2}\)

3. Determine the gradient of the perpendicular bisector:

\(m_{\text{perp}} = -\frac{1}{m} = 2\)

4. Write the equation of the perpendicular bisector using point-slope form:

\(y - 5 = 2(x - 6)\)

5. Set \(y = 0\) to find the x-intercept:

\(0 - 5 = 2(x - 6)\)

\(-5 = 2x - 12\)

\(2x = 7\)

\(x = \frac{7}{2}\)

6. The coordinates where the perpendicular bisector meets the x-axis are:

\(\left( \frac{7}{2}, 0 \right)\)