Straight LineHardBloom L3
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:
Options
A.$2m(m + n)$
B.$m(m + 3n)$
C.$m(2m + 3n)$
D.$2m(m + 3n)$
Solution
Sol. [Key Concept: Successive reflections of $(m,n)$ across $y = x$ and the axes form a pentagon; area $=$ rectangle $+$ triangle.]
Area of rectangle $BCDE = 4mn$
$$\text{~}\text{Area o}\text{f}$$
$$\text{~}\begin{matrix} \bigtriangleup ABE & \ = \frac{2m(m - n)}{2} \\ & \ = m^{2} - mn \end{matrix}$$
∴ area of pentagon $= 4mn + m^{2} - mn = m^{2} + 3mn$
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