We have, $\cos \frac{B}{2}=\frac{\frac{9}{2}+4-1}{6 \sqrt{2}}=\frac{5}{4 \sqrt{2}}$
$\therefore$ Length of angle bisector,
$B D=\frac{2 a c}{a+c} \cos \frac{B}{2}$
$\frac{3}{\sqrt{2}} =\left(\frac{4 c}{e+2}\right) \cdot \frac{5}{4 \sqrt{2}}$
$c =3$
We know that, $\frac{A B}{B C}=\frac{A D}{C D}$
$A D=\frac{3}{2}$
$\therefore$ Perimeter of $\triangle A B C=1+\frac{3}{2}+3+2=\frac{15}{2}$
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$\mathrm{A}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}:(\mathrm{x}-2)^{2}+\mathrm{y}^{2} \leq 4\right\}$
$\mathrm{B}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}: \mathrm{x}^{2}+\mathrm{y}^{2} \leq 4\right\} \text { and }$
$\mathrm{C}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{Z} \times \mathbb{Z}:(\mathrm{x}-2)^{2}+(\mathrm{y}-2)^{2} \leq 4\right\}$
If the total number of relation from $\mathrm{A} \cap \mathrm{B}$ to $\mathrm{A} \cap \mathrm{C}$ is $2^{\mathrm{p}}$, then the value of $\mathrm{p}$ is :