Answer:
24 cm
Step-by-step explanation:
Since, both the Pentagons are similar. So, their corresponding sides would be in proportion.
Therefore,
[tex] \frac{4}{4x} = \frac{(x + 2)}{48} \\ \\ \frac{1}{x} = \frac{(x + 2)}{48} \\ \\ x(x + 2) = 48 \\ \\ {x}^{2} + 2x = 48 \\ \\ {x}^{2} + 2x - 48 = 0 \\ \\ {x}^{2} + 8x - 6x - 48 = 0 \\ \\ x(x + 8) - 6(x + 8) = 0 \\ \\ (x + 8)(x - 6) = 0 \\ \\ x + 8 = 0 \: \: or \: \: x - 6 = 0 \\ \\ x = - 8 \: \: or \: \: x = 6 \\ \\ \because \: x \: cannot \: be \: - ve \\ \implies \: x = 6[/tex]
Width of badge = 4x = 4*6 = 24 cm
