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: Understanding the problem:
We are asked to find the point where the perpendicular bisector of the line segment joining the points A(–1, 3) and B(2, 4) cuts the y-axis.
To solve this, we need to follow these steps:
1. Find the midpoint of the line segment AB.
2. Find the slope of the line segment AB.
3. Determine the slope of the perpendicular bisector (it will be the negative reciprocal of the slope of AB).
4. Use the point-slope form of the equation of a line to find the equation of the perpendicular bisector.
5. Find the y-intercept by setting \( x = 0 \).

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

Step 3: Finding the slope of line AB:
The slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ \text{slope of AB} = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the coordinates of A(–1, 3) and B(2, 4):
\[ \text{slope of AB} = \frac{4 - 3}{2 - (-1)} = \frac{1}{3} \]

Step 4: Finding the slope of the perpendicular bisector:
The slope of the perpendicular bisector is the negative reciprocal of the slope of AB.
Thus, the slope of the perpendicular bisector is:
\[ \text{slope of perpendicular bisector} = -\frac{3}{1} = -3 \]

Step 5: Finding the equation of the perpendicular bisector:
The perpendicular bisector passes through the midpoint \( M \left( \frac{1}{2}, \frac{7}{2} \right) \), and its slope is -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 \( y_1 = \frac{7}{2} \):
\[ y - \frac{7}{2} = -3 \left( x - \frac{1}{2} \right) \] Simplifying:
\[ y - \frac{7}{2} = -3x + \frac{3}{2} \] \[ y = -3x + \frac{3}{2} + \frac{7}{2} \] \[ y = -3x + \frac{10}{2} = -3x + 5 \]

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

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