To find the value of \( b \) in the rectangle \( EFGH \), we need to establish the relationships from the geometry of the rectangle.
Given that:
- \( GI = b + 54 \)
- \( HI = 19b \)
In a rectangle, the opposite sides are equal. Thus:
- \( GI \) is a side of the rectangle.
- \( HI \) is the adjacent side of the rectangle.
In the context of a rectangle, since both are lengths, we can equate the values of \( GI \) and \( HI \) under the conditions that they are the sides of the rectangle:
\[ b + 54 = 19b \]
Now, solve this equation for \( b \):
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Subtract \( b \) from both sides: \[ 54 = 19b - b \] \[ 54 = 18b \]
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Now divide both sides by 18: \[ b = \frac{54}{18} \] \[ b = 3 \]
Thus, the value of \( b \) is:
\[ \boxed{3} \]