If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.
\(x = 6,\ y = 3\)
Let the vertices of the parallelogram taken in order be \(A(1, 2)\), \(B(4, y)\), \(C(x, 6)\) and \(D(3, 5)\).
Property used: The diagonals of a parallelogram bisect each other. Therefore, the midpoint of diagonal \(AC\) is equal to the midpoint of diagonal \(BD\).
Midpoint formula: The midpoint of the line segment joining \((x_1, y_1)\) and \((x_2, y_2)\) is given by
\( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
Step 1: Find the midpoint of diagonal \(AC\)
Coordinates of \(A\) and \(C\) are \((1, 2)\) and \((x, 6)\).
Midpoint of \(AC\) = \( \left( \frac{1 + x}{2}, \frac{2 + 6}{2} \right) = \left( \frac{1 + x}{2}, 4 \right) \).
Step 2: Find the midpoint of diagonal \(BD\)
Coordinates of \(B\) and \(D\) are \((4, y)\) and \((3, 5)\).
Midpoint of \(BD\) = \( \left( \frac{4 + 3}{2}, \frac{y + 5}{2} \right) = \left( \frac{7}{2}, \frac{y + 5}{2} \right) \).
Step 3: Equate the midpoints
Since the diagonals bisect each other, the corresponding coordinates of the midpoints are equal.
Equating x-coordinates:
\( \frac{1 + x}{2} = \frac{7}{2} \Rightarrow 1 + x = 7 \Rightarrow x = 6 \).
Equating y-coordinates:
\( 4 = \frac{y + 5}{2} \Rightarrow 8 = y + 5 \Rightarrow y = 3 \).
Conclusion: The values of the variables are \(x = 6\) and \(y = 3\).