Consider a point A(m,n) , where m and n are positve intergers. B is the reflection of A in the line `y=x`, C is the reflaction of B in the y axis, D is the reflection of C in the x axis and E is the reflection of D is the y axis. The area of the pentagon ABCDE is a. `2m (m+n)` b. `m (m+3n)` c. `m(2m+3n)` d. `2m(m+3n)`

Options

A. `2m (m+n)`
B. `m (m+3n)`
C. `m(2m+3n)`
D. `2m(m+3n)`

Explanation

To solve the problem, we will follow the steps outlined in the video transcript and derive the area of the pentagon ABCDE.

### Step-by-Step Solution:

1. **Identify the Coordinates of Point A**:
- Let point A be \( A(m, n) \), where \( m \) and \( n \) are positive integers.

2. **Find the Coordinates of Point B**:
- Point B is the reflection of A across the line \( y = x \).
- The coordinates of point B will be \( B(n, m) \).

3. **Find the Coordinates of Point C**:
- Point C is the reflection of B across the y-axis.
- The coordinates of point C will be \( C(-n, m) \).

4. **Find the Coordinates of Point D**:
- Point D is the reflection of C across the x-axis.
- The coordinates of point D will be \( D(-n, -m) \).

5. **Find the Coordinates of Point E**:
- Point E is the reflection of D across the y-axis.
- The coordinates of point E will be \( E(n, -m) \).

6. **Plot the Points**:
- The coordinates of the points are:
- \( A(m, n) \)
- \( B(n, m) \)
- \( C(-n, m) \)
- \( D(-n, -m) \)
- \( E(n, -m) \)

7. **Divide the Pentagon into Shapes**:
- The pentagon ABCDE can be divided into a rectangle and a triangle.
- The rectangle is formed by points A, B, D, and E.
- The triangle is formed by points A, C, and D.

8. **Calculate the Area of the Rectangle**:
- The length of the rectangle is \( 2m \) (from A to E).
- The height of the rectangle is \( 2n \) (from A to D).
- Area of the rectangle = Length × Height = \( 2m \times 2n = 4mn \).

9. **Calculate the Area of the Triangle**:
- The base of the triangle (AC) is \( 2m \) (from A to C).
- The height of the triangle is \( m - n \) (vertical distance from A to C).
- Area of the triangle = \( \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 2m \times (m - n) = m(m - n) \).

10. **Total Area of Pentagon ABCDE**:
- Total Area = Area of Rectangle + Area of Triangle
- Total Area = \( 4mn + m(m - n) = 4mn + m^2 - mn = m^2 + 3mn \).

11. **Final Expression**:
- The total area can be factored as \( m(m + 3n) \).

### Conclusion:
The area of the pentagon ABCDE is given by \( m(m + 3n) \).