To find the value of \( x \) for the isosceles triangle, we start with the information provided.
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Identify the sides of the triangle:
- The base of the triangle is \( x \).
- Each leg of the triangle is \( 22x - 77 \).
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Set up the perimeter equation: The perimeter \( P \) of a triangle is the sum of the lengths of all its sides. Therefore, we can write:
\[ \text{Perimeter} = \text{base} + 2 \times \text{leg} \]
Substituting the expressions we have:
\[ 2121 = x + 2(22x - 77) \]
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Simplify the equation: First, distribute the 2:
\[ 2121 = x + 44x - 154 \]
Combine like terms:
\[ 2121 = 45x - 154 \]
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Isolate the variable \( x \): Add 154 to both sides:
\[ 2121 + 154 = 45x \]
\[ 2275 = 45x \]
Divide both sides by 45:
\[ x = \frac{2275}{45} \]
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Simplify the fraction: To simplify \( \frac{2275}{45} \), we can divide both the numerator and the denominator by their greatest common divisor.
After simplifying, we find:
\[ x = 50.55556 \text{ or } x \approx 50.56 \]
Thus, the final answer for \( x \) is:
\[ \boxed{50.56} \]