To find the value of '\(a\)' for the parallelogram with vertices \(A(1, 2)\), \(B(a, -26)\), \(C(13, -14)\), and \(D(6, 14)\) taken in order, we use the property that the diagonals of a parallelogram bisect each other.

Parallelogram Vertices Property

The midpoint of diagonal \(AC\) must be the same as the midpoint of diagonal \(BD\). The midpoint formula for two points \((x_1, y_1)\) and \((x_2, y_2)\) is \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\).

Calculate Diagonal Midpoints

• Midpoint of AC: Using points \(A(1, 2)\) and \(C(13, -14)\):

  Midpoint\(_{AC} = (\frac{1+13}{2}, \frac{2+(-14)}{2}) = (\frac{14}{2}, \frac{-12}{2}) = (7, -6)\)

• Midpoint of BD: Using points \(B(a, -26)\) and \(D(6, 14)\):

  Midpoint\(_{BD} = (\frac{a+6}{2}, \frac{-26+14}{2}) = (\frac{a+6}{2}, \frac{-12}{2}) = (\frac{a+6}{2}, -6)\)

Solve for Unknown Coordinate 'a'

Equating the x-coordinates of the midpoints:

\(7 = \frac{a+6}{2}\)

Multiply both sides by 2:

\(7 \times 2 = a+6\)

\(14 = a+6\)

Subtract 6 from both sides:

\(a = 14 - 6\)

\(a = 8\)

Thus, the value of '\(a\)' is 8.