Quadrilateral EFGH is a rectangle, GI=b+54, and HI=19b. What is the value of b?

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1 answer

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:

  1. \( GI \) is a side of the rectangle.
  2. \( HI \) 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 \( b \):

  1. Subtract \( b \) from both sides: \[ 54 = 19b - b \] \[ 54 = 18b \]

  2. Now divide both sides by 18: \[ b = \frac{54}{18} \] \[ b = 3 \]

Thus, the value of \( b \) is:

\[ \boxed{3} \]