First find the mid point
x coordinate = 1+4 / 2 = 2.5
y coordinate = 5+6 / 2 = 5.5
Mid point = (2.5, 5.5)

Then find the slope of the bisector :
Slope of the given line = (5-6) / (1-4) = 1/3
Slope of given line multiplied by slope of bisector = -1
Slope of bisector = -1 / (1/3)
= -3

Use the point slope form to find the bisector's formula :
-3 = (5.5 - y) / (2.5 - x)
-7.5 + 3x = 5.5 - y
3x + y - 13 = 0

Transform the formula into slope-intercept form

3x + y - 13 = 0
y = -3x + 13

Because slope-intercept form is y = mx + c, where m is the slope and c is the y-intercept

Therefore the perpendicular bisector cuts the y-axis at (0,13)

To find the point where the perpendicular bisector of the line segment joining points A(2, 5) and B(4, 7) cuts, we can follow these steps:

1. Find the midpoint of the line segment AB:
- The midpoint formula is given by:
((x1 + x2) / 2, (y1 + y2) / 2)
- Substituting the coordinates of A and B:
Midpoint = ((2 + 4) / 2, (5 + 7) / 2)
= (6 / 2, 12 / 2)
= (3, 6)

2. Find the slope of the line segment AB:
- The slope formula is given by:
(y2 - y1) / (x2 - x1)
- Substituting the coordinates of A and B:
Slope = (7 - 5) / (4 - 2)
= 2 / 2
= 1

3. Find the negative reciprocal of the slope:
- The negative reciprocal of a slope is obtained by flipping the fraction and changing the sign.
- In this case, the negative reciprocal of 1 is -1.

4. Find the equation of the perpendicular bisector:
- The equation of a line in point-slope form is given by:
y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
- Using the midpoint (3, 6) as a point on the line and the negative reciprocal slope (-1):
y - 6 = -1(x - 3)
y - 6 = -x + 3
y = -x + 3 + 6
y = -x + 9

5. Find the intersection point of the perpendicular bisector with the x and y axes:
- To find the intersection point with the x-axis, we set y = 0 in the equation of the line and solve for x:
0 = -x + 9
x = 9
- Therefore, the intersection point with the x-axis is (9, 0).
- To find the intersection point with the y-axis, we set x = 0 in the equation of the line and solve for y:
y = -0 + 9
y = 9
- Therefore, the intersection point with the y-axis is (0, 9).

6. Conclusion:
- The point where the perpendicular bisector of the line segment AB cuts is (9, 0), which corresponds to option B.