Question

$m,n$ are integers with $0<n<m.A$ is the point $(m,n)$ on the Cartesian plane. $B$ is the reflection of $A$ in the line $y=x.C$ is the reflection of $B$ in the $y-$axis, $D$ is the reflection of $C$ in the $x-$axis and $E$ is the reflection of $D$ in the $y-$axis. The area of the pentagon $ABCDE$ is

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

Answer 1

The correct option is B $m(m+3n)$

Image of $A(m,n)$ in $y=x$ is $B(n,m)$
Image of $B(n,m)$ in $y-axis$ is $C(-n,m)$
Image of $C(-n,m)$ in $x-axis$ is $D(-n,-m)$
Image of $D(-n,-m)$ in $y-axis$ is $E(n,-m)$
Area of rectangle $BCDE$ is $2m\times 2n=4mn$
Area of $△ABE=\frac{1}{2}\times 2m\times (m-n)=m^2-mn$
Total area $=4mn+m^2-mn=3mn+m^2$