Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
$m, n$ are integer 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:
$2m(m+n)$
$m(m+3n)$
$m(2m+3n)$
$2m(m+3n)$
Step-by-Step Solution
Key Concept: Decompose composite geometric figures into simpler shapes (rectangles and triangles) whose areas can be computed separately and summed.
The rectangle $BCDE$ has area $4mn$. Triangle $ABE$ has base $2n$ and height $2(m-n)$, so area equals $m^2 - mn$. The pentagon is formed by combining regions, giving total area $4mn + m^2 - mn = m^2 + 3mn$.
Correct Answer: 2