If the points A (1,-2), B (2,3) C (a,2) and D (-4,3) from a parallelogram find the value of a and height of the parallelogram taking AB as base.

Answer

To solve the problem, we need to find the value of \( a \) and the height of the parallelogram formed by the points \( A(1, -2) \), \( B(2, 3) \), \( C(a, 2) \), and \( D(-4, 3) \), taking \( AB \) as the base.

### Step 1: Find the coordinates of points A and B
The coordinates of point \( A \) are \( (1, -2) \) and the coordinates of point \( B \) are \( (2, 3) \).

### Step 2: Calculate the length of base AB
The length of line segment \( AB \) can be calculated using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points \( A \) and \( B \):
\[ AB = \sqrt{(2 - 1)^2 + (3 - (-2))^2} = \sqrt{(1)^2 + (5)^2} = \sqrt{1 + 25} = \sqrt{26} \]

### Step 3: Find the coordinates of point D
The coordinates of point \( D \) are \( (-4, 3) \).

### Step 4: Find the height of the parallelogram
To find the height of the parallelogram, we need to find the perpendicular distance from point \( C(a, 2) \) to the line \( AB \).

### Step 5: Find the equation of line AB
The slope \( m \) of line \( AB \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - (-2)}{2 - 1} = \frac{5}{1} = 5 \]
Using point-slope form, the equation of line \( AB \) can be written as:
\[ y - y_1 = m(x - x_1) \implies y - (-2) = 5(x - 1) \implies y + 2 = 5x - 5 \implies y = 5x - 7 \]

### Step 6: Find the perpendicular distance from point C to line AB
The formula for the distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Rearranging the equation of line \( AB \) to standard form:
\[ 5x - y - 7 = 0 \implies A = 5, B = -1, C = -7 \]
Substituting the coordinates of point \( C(a, 2) \):
\[ d = \frac{|5a - 1(2) - 7|}{\sqrt{5^2 + (-1)^2}} = \frac{|5a - 2 - 7|}{\sqrt{25 + 1}} = \frac{|5a - 9|}{\sqrt{26}} \]

### Step 7: Find the value of a
Since \( C \) is the opposite vertex of the parallelogram, the height from \( C \) to line \( AB \) must equal the height from \( D \) to line \( AB \). The coordinates of point \( D \) are \( (-4, 3) \). We can find the height from \( D \) to line \( AB \):
\[ d = \frac{|5(-4) - 1(3) - 7|}{\sqrt{26}} = \frac{|-20 - 3 - 7|}{\sqrt{26}} = \frac{|-30|}{\sqrt{26}} = \frac{30}{\sqrt{26}} \]
Setting the heights equal gives:
\[ \frac{|5a - 9|}{\sqrt{26}} = \frac{30}{\sqrt{26}} \]
Thus, we have:
\[ |5a - 9| = 30 \]
This leads to two equations:
1. \( 5a - 9 = 30 \)
2. \( 5a - 9 = -30 \)

Solving the first equation:
\[ 5a = 39 \implies a = \frac{39}{5} = 7.8 \]
Solving the second equation:
\[ 5a = -21 \implies a = -\frac{21}{5} = -4.2 \]

### Step 8: Conclusion
The possible values for \( a \) are \( 7.8 \) and \( -4.2 \). The height of the parallelogram is \( \frac{30}{\sqrt{26}} \).

### Final Answer
- The value of \( a \) can be \( 7.8 \) or \( -4.2 \).
- The height of the parallelogram is \( \frac{30}{\sqrt{26}} \).