If (1,2), (4,y), (x,6), and (3,5) are the vertices of a parallelogram taken in order, find x and y.
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Step-by-Step Solution
Step 1: Understand Parallelogram Properties
A key property of a parallelogram is that its diagonals bisect each other. This means that the midpoint of one diagonal is the same as the midpoint of the other diagonal. We will use the midpoint formula to set up equations for x and y.
Step 2: Calculate Midpoint of Diagonal AC
Let the given vertices be A(1,2), B(4,y), C(x,6), and D(3,5). The diagonals are AC and BD. First, we calculate the midpoint of diagonal AC using the coordinates of A and C.
Step 3: Calculate Midpoint of Diagonal BD
Next, we calculate the midpoint of the other diagonal, BD, using the coordinates of B and D. This will give us an expression involving y.
Step 4: Equate Midpoints and Solve for x and y
Since the midpoints of the diagonals are the same, we can equate their respective x-coordinates and y-coordinates. This gives us two simple equations to solve for x and y.