To find the value of \( x \) for the given isosceles triangle, let's denote the lengths as follows:
- The length of the base \( b = x \).
- The length of each leg \( l = 22x - 77 \).
Since the triangle is isosceles, the perimeter \( P \) can be expressed as:
\[ P = b + 2l \]
Substituting the expressions for \( b \) and \( l \) into the perimeter equation:
\[ 2121 = x + 2(22x - 77) \]
Now, simplify the equation:
-
Distribute \( 2 \) to \( (22x - 77) \): \[ 2121 = x + 44x - 154 \]
-
Combine like terms: \[ 2121 = 45x - 154 \]
-
Add \( 154 \) to both sides: \[ 2121 + 154 = 45x \] \[ 2275 = 45x \]
-
Divide by \( 45 \): \[ x = \frac{2275}{45} \]
Now, simplify \( \frac{2275}{45} \). Calculate it efficiently:
- Find a common factor (if any):
- Divide the numerator and denominator by \( 5 \): \[ x = \frac{455}{9} \]
Thus, the value of \( x \) is:
\[ x = \frac{455}{9} \]
This gives us the result, and we can also evaluate \( x \) as a decimal for practical purposes:
\[ x \approx 50.56 \]
Thus, the final answer for \( x \) is:
\[ \boxed{\frac{455}{9}} \] or approximately \( \boxed{50.56} \).