Question

Find the equation of the right bisector of the line segment joining the points \( (3,4) \) and \( (1,2) \).

Hint:

The right bisector of a line segment passes through its midpoint and has a slope that is the negative reciprocal of the slope of the line segment.

Answer 1

Step 1: Find the midpoint of the line segment.
The midpoint \( M \) of the line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substitute the coordinates of the points \( (3, 4) \) and \( (1, 2) \): \[ M = \left( \frac{3 + 1}{2}, \frac{4 + 2}{2} \right) = \left( \frac{4}{2}, \frac{6}{2} \right) = (2, 3) \]
Step 2: Find the slope of the line segment.}
The slope \( m \) of the line joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the coordinates \( (3, 4) \) and \( (1, 2) \): \[ m = \frac{2 - 4}{1 - 3} = \frac{-2}{-2} = 1 \]
Step 3: Find the slope of the perpendicular bisector.}
The slope of the right bisector (perpendicular to the line) is the negative reciprocal of the slope of the original line. Therefore, the slope of the bisector is: \[ m_{\text{bisector}} = -\frac{1}{1} = -1 \]
Step 4: Use the point-slope form of the equation.}
The equation of a line with slope \( m \) passing through a point \( (x_1, y_1) \) is given by: \[ y - y_1 = m(x - x_1) \] Substitute the slope \( m = -1 \) and the point \( (2, 3) \) into the equation: \[ y - 3 = -1(x - 2) \] Simplify the equation: \[ y - 3 = -x + 2 \] \[ y = -x + 5 \] So, the equation of the right bisector is \( y = -x + 5 \).