Question

Find the equation of the right bisector of the line segment joining the points (1, 2) and (5, 6).

Answer 1

The midpoint M of the segment joining A(1, 2) and B(5, 6) is calculated using the midpoint formula:

`M = ((x_1 + x_2)/2, (y_1 + y_2)/2)`

`M = (1 + 5)/2, (2 + 6)/2`

`M = (6/2, 8/2)`

∴ M = (3, 4)

The slope (m₁) of the line joining (1, 2) and (5, 6) is:

`m_1 = (y_2 - y_1)/(x_2 - x_1)`

`m_1 = (6 - 2)/(5 - 1)`

`m_1 = 4/4`

∴ m₁ = 1

Since the right bisector is perpendicular to the segment, the product of their slopes is −1,

Let m₂ be the slope of the bisector:

m₁ × m₂ = −1

1 × m₂ = −1

∴ m₂ = −1

Using the point-slope formula with the midpoint (3, 4) and slope m₂ = −1:

y − y₁​ = m(x − x₁​)

y − 4 = −1(x − 3)

y − 4 = −x + 3

Let’s rearrange into the general form (Ax + By + C = 0):

x + y − 4 − 3 = 0

x + y − 7 = 0

Hence, the equation of the right bisector of the line segment joining the points (1, 2) and (5, 6) is x + y − 7 = 0.