Question

What are the coordinates of the point where the perpendicular bisector of the line segment joining the points A (1, 5), and B (4, 6) cuts the y-axis?

Answer 1

Given: A(1, 5) and B(4, 6).

Step-wise calculation:

1. Midpoint P of AB:

`P = ((1 + 4)/2, (5 + 6)/2)` 

= `(5/2, 11/2)`

2. Slope of AB:

`m_(AB) = (6 - 5)/(4 - 1)` 

= `1/3`

3. Slope of perpendicular bisector:

`m_"perp" = -1/m_(AB)`

= –3

4. Equation of perpendicular bisector through `P(5/2, 11/2)`:

`y - 11/2 = -3(x - 5/2)` 

⇒ `y − 11/2`

= `-3x + 15/2` 

⇒ `y = -3x + (15/2 + 11/2)`

= –3x + 13

5. Intersection with the y-axis (x = 0):

y = –3(0) + 13

= 13

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