Question

The perpendicular bisector of the line segment joining the points A(–1, 3) and B(2, 4) cuts the y-axis at :

• A. (0, 5)
• B. (0, -5)
• C. (0, 4)
• D. (0, -4)

Answer 1

The Correct Option is A

Step 1: Understand the problem:
We are given two points \( A(-1, 3) \) and \( B(2, 4) \), and we need to find the point where the perpendicular bisector of the line segment joining \( A \) and \( B \) cuts the y-axis.

Step 2: Find the midpoint of the line segment joining A and B:
The midpoint \( M \) of a line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by the formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substitute the coordinates of points \( A(-1, 3) \) and \( B(2, 4) \): \[ M = \left( \frac{-1 + 2}{2}, \frac{3 + 4}{2} \right) = \left( \frac{1}{2}, \frac{7}{2} \right) \] So, the midpoint is \( M\left( \frac{1}{2}, \frac{7}{2} \right) \).

Step 3: Find the slope of the line segment AB:
The slope \( m \) of the line joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the coordinates of points \( A(-1, 3) \) and \( B(2, 4) \): \[ m = \frac{4 - 3}{2 - (-1)} = \frac{1}{3} \] So, the slope of line \( AB \) is \( \frac{1}{3} \).

Step 4: Find the slope of the perpendicular bisector:
The slope of the perpendicular bisector is the negative reciprocal of the slope of line \( AB \). Therefore, the slope of the perpendicular bisector is: \[ m_{\text{perpendicular}} = -\frac{1}{\frac{1}{3}} = -3 \]

Step 5: Find the equation of the perpendicular bisector:
We know that the perpendicular bisector passes through the midpoint \( M\left( \frac{1}{2}, \frac{7}{2} \right) \) and has a slope of \( -3 \). Using the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \] Substitute \( m = -3 \), \( x_1 = \frac{1}{2} \), and \( y_1 = \frac{7}{2} \): \[ y - \frac{7}{2} = -3\left( x - \frac{1}{2} \right) \] Simplify the equation: \[ y - \frac{7}{2} = -3x + \frac{3}{2} \] Add \( \frac{7}{2} \) to both sides: \[ y = -3x + \frac{3}{2} + \frac{7}{2} = -3x + \frac{10}{2} = -3x + 5 \]

Step 6: Find the point where the perpendicular bisector cuts the y-axis:
The perpendicular bisector cuts the y-axis where \( x = 0 \). Substitute \( x = 0 \) into the equation of the perpendicular bisector: \[ y = -3(0) + 5 = 5 \] Thus, the perpendicular bisector cuts the y-axis at the point \( (0, 5) \).

Conclusion:
The perpendicular bisector of the line segment joining points \( A(-1, 3) \) and \( B(2, 4) \) cuts the y-axis at \( \boxed{(0, 5)} \).