Point P (0, - 7) is the point of intersection of y-axis and perpendicular bisector of line segment joining the points A (-1, 0) and B (7, -6).

Answer

To determine whether point P (0, -7) is the point of intersection of the y-axis and the perpendicular bisector of the line segment joining points A (-1, 0) and B (7, -6), we need to verify two conditions:

1. **Check if point P lies on the y-axis.**
2. **Check if point P is the midpoint of line segment AB.**

### Step 1: Verify if Point P lies on the y-axis

The coordinates of point P are (0, -7). A point lies on the y-axis if its x-coordinate is 0.

- Since the x-coordinate of P is 0, we can conclude that point P lies on the y-axis.

### Step 2: Find the midpoint of line segment AB

The coordinates of points A and B are A (-1, 0) and B (7, -6). The formula for the midpoint M of a line segment joining two points (x1, y1) and (x2, y2) is given by:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Substituting the coordinates of points A and B:

\[ M = \left( \frac{-1 + 7}{2}, \frac{0 + (-6)}{2} \right) \]

Calculating the x-coordinate:

\[ \frac{-1 + 7}{2} = \frac{6}{2} = 3 \]

Calculating the y-coordinate:

\[ \frac{0 - 6}{2} = \frac{-6}{2} = -3 \]

Thus, the midpoint M of line segment AB is:

\[ M = (3, -3) \]

### Step 3: Check if Point P is the midpoint

Point P has coordinates (0, -7) and the midpoint M has coordinates (3, -3). Since (0, -7) is not equal to (3, -3), point P is not the midpoint of line segment AB.

### Conclusion

- Point P (0, -7) lies on the y-axis (True).
- Point P (0, -7) is not the midpoint of line segment AB (False).

Therefore, the statement that point P is the point of intersection of the y-axis and the perpendicular bisector of line segment AB is **not true**.

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