Given points `A(1, 5), B(-3,7) and C(15,9)`
(i) find the equation of the line passing through the mid point of AC and point B
(ii) find the equation of the line through C and parallel to AB
(iii)The lines obtained in parts (i) and (ii) above, intersect each other at a point P. Find the co-ordinates of the point P.
(iv) assign, giving reason, a special names of the figure PABC

Answer

Let's solve the problem step by step.

### Given Points:
- A(1, 5)
- B(-3, 7)
- C(15, 9)

### (i) Find the equation of the line passing through the midpoint of AC and point B.

1. **Find the Midpoint of AC:**
The midpoint \( M \) of points \( A(x_1, y_1) \) and \( C(x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of A and C:
\[ M = \left( \frac{1 + 15}{2}, \frac{5 + 9}{2} \right) = \left( \frac{16}{2}, \frac{14}{2} \right) = (8, 7) \]

2. **Use the Two Points to Find the Equation of the Line:**
We need the equation of the line passing through points \( M(8, 7) \) and \( B(-3, 7) \).
The slope \( m \) of the line through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \( y_1 = 7 \) and \( y_2 = 7 \) (both points have the same y-coordinate), so:
\[ m = \frac{7 - 7}{-3 - 8} = \frac{0}{-11} = 0 \]
Since the slope is 0, the line is horizontal. The equation of the line is:
\[ y = 7 \]

### (ii) Find the equation of the line through C and parallel to AB.

1. **Find the Slope of Line AB:**
Using points A(1, 5) and B(-3, 7):
\[ m_{AB} = \frac{7 - 5}{-3 - 1} = \frac{2}{-4} = -\frac{1}{2} \]

2. **Equation of Line through C(15, 9) with the same slope:**
Using the point-slope form of the line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = -\frac{1}{2} \) and point C(15, 9):
\[ y - 9 = -\frac{1}{2}(x - 15) \]
Simplifying:
\[ y - 9 = -\frac{1}{2}x + \frac{15}{2} \]
\[ y = -\frac{1}{2}x + \frac{15}{2} + 9 \]
\[ y = -\frac{1}{2}x + \frac{15}{2} + \frac{18}{2} = -\frac{1}{2}x + \frac{33}{2} \]

### (iii) Find the intersection point P of the two lines.

1. **Set the equations equal to find P:**
From part (i), we have \( y = 7 \).
From part (ii), we have:
\[ 7 = -\frac{1}{2}x + \frac{33}{2} \]

2. **Solve for x:**
\[ 7 = -\frac{1}{2}x + 16.5 \]
\[ -\frac{1}{2}x = 7 - 16.5 \]
\[ -\frac{1}{2}x = -9.5 \]
\[ x = 19 \]

3. **Coordinates of Point P:**
Therefore, the coordinates of point P are \( (19, 7) \).

### (iv) Assign a special name to the figure PABC.

The figure PABC is a **parallelogram** because:
- The line segment AB is parallel to the line segment PC (as they have the same slope).
- The line segment AC is parallel to the line segment PB (as they have the same slope).

### Summary of Solutions:
1. Equation of line through midpoint of AC and B: \( y = 7 \)
2. Equation of line through C and parallel to AB: \( y = -\frac{1}{2}x + \frac{33}{2} \)
3. Intersection point P: \( (19, 7) \)
4. Figure PABC is a parallelogram.