The perpendicular bisector of the line segment joining the points A(1,5) and B(4,6) cuts the Y-axis at

Options

A. (0,13)
B. (0,-13)
C. (0,12)
D. (13,0)

Explanation

To find the point where the perpendicular bisector of the line segment joining the points A(1,5) and B(4,6) cuts the Y-axis, we can follow these steps:

### Step 1: Find the Midpoint of AB
The midpoint \( M \) of the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
For points \( A(1, 5) \) and \( B(4, 6) \):
\[ M = \left( \frac{1 + 4}{2}, \frac{5 + 6}{2} \right) = \left( \frac{5}{2}, \frac{11}{2} \right) = \left( 2.5, 5.5 \right) \]

### Step 2: Find the Slope of AB
The slope \( m \) of the line segment joining points \( A \) and \( B \) is calculated as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points \( A \) and \( B \):
\[ m = \frac{6 - 5}{4 - 1} = \frac{1}{3} \]

### Step 3: Find the Slope of the Perpendicular Bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of \( AB \):
\[ m_{perpendicular} = -\frac{1}{m} = -3 \]

### Step 4: Write the Equation of the Perpendicular Bisector
Using the point-slope form of the equation of a line, which is \( y - y_1 = m(x - x_1) \), we can write the equation of the perpendicular bisector using the midpoint \( M(2.5, 5.5) \):
\[ y - 5.5 = -3(x - 2.5) \]
Expanding this:
\[ y - 5.5 = -3x + 7.5 \]
\[ y = -3x + 13 \]

### Step 5: Find the Intersection with the Y-axis
To find where this line intersects the Y-axis, we set \( x = 0 \):
\[ y = -3(0) + 13 = 13 \]
Thus, the point where the perpendicular bisector cuts the Y-axis is \( (0, 13) \).

### Final Answer
The perpendicular bisector of the line segment joining points A(1,5) and B(4,6) cuts the Y-axis at the point \( (0, 13) \).

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