Question 1125382
if they're similar, then their sides are proportional.


that gets you x / 4x = (x + 2) / 48


cross multiply to get 48x = 4x * (x + 2)


simplify to get 48x = 4x^2 + 8x


subtract 48x from both sides to get 0 = 4x^2 + 8x - 48x


combine like terms to get 0 = 4x^2 - 40x


factor to get 0 = 4x * (x - 10)


solve for x to get x = 0 or x = 10


x = 0 is not good, so it has to be x = 10 or not at all.


when x = 10, the smaller figure is 10 by 12 and the larger figure is 40 by 48.


the sides are proportional, as they should be with similar figures.


10 / 40 = 12 / 48


cross multiply to get 480 = 480, confirming the ratios are equal which confirms that the sides are proportional.


your solution is that the width of the badge is 40 cm.


that's the larger pentagon that has the smaller pentagon as part of it, if i understand the problem correctly.