The point which lies on the perpendicular bisector of the line segment joining the points A(-2,-5) and B (2,5) is

Options

A. (0,0)
B. (0,2)
C. (2,0)
D. (-2,0)

Explanation

To find the point that lies on the perpendicular bisector of the line segment joining the points A(-2, -5) and B(2, 5), we will follow these steps:

### Step 1: Identify the Coordinates of Points A and B
The coordinates of point A are (-2, -5) and the coordinates of point B are (2, 5).

### Step 2: Calculate the Midpoint of Line Segment AB
The formula for the midpoint \( C(x, y) \) of a line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[ C = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, \( x_1 = -2, y_1 = -5, x_2 = 2, y_2 = 5 \).

Substituting the values:
\[ C_x = \frac{-2 + 2}{2} = \frac{0}{2} = 0 \]
\[ C_y = \frac{-5 + 5}{2} = \frac{0}{2} = 0 \]
Thus, the coordinates of the midpoint \( C \) are \( (0, 0) \).

### Step 3: Determine the Slope of AB
Next, we find the slope of line segment AB. The slope \( m \) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points A and B:
\[ m = \frac{5 - (-5)}{2 - (-2)} = \frac{5 + 5}{2 + 2} = \frac{10}{4} = \frac{5}{2} \]

### Step 4: Calculate the Slope of the Perpendicular Bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of AB. Therefore:
\[ m_{perpendicular} = -\frac{1}{m} = -\frac{1}{\frac{5}{2}} = -\frac{2}{5} \]

### Step 5: Equation of the Perpendicular Bisector
The equation of a line in point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
Using the midpoint \( C(0, 0) \) and the slope \( -\frac{2}{5} \):
\[ y - 0 = -\frac{2}{5}(x - 0) \]
This simplifies to:
\[ y = -\frac{2}{5}x \]

### Conclusion
The point that lies on the perpendicular bisector of the line segment joining points A and B is any point that satisfies the equation \( y = -\frac{2}{5}x \). A simple example is the origin \( (0, 0) \), which we already calculated as the midpoint.