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If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find the values of x and y.

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x = 11, y = 10

x = 7, y = 8

x = 6, y = 3

x = 4, y = 5

answer is C.

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Detailed Solution

Given that (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram.
Question Image

So, the coordinates are A(1,2), B(4, y), C(x, 6), and D(3,5).
We know that diagonals of parallelograms bisect each other.
So, the midpoint of AC is the same as the midpoint of BD.
Thus, by midpoint formula,

(\frac{x_{3} + x_{1}}{2}, \frac{y_{3} + y_{1}}{2}) = (\frac{x_{4} + x_{2}}{2}, \frac{y_{4} + y_{2}}{2})

Here,

{(x}_{1}, y_{1}) = (1, 2), {(x}_{2}, y_{2}) = (4, y), {(x}_{3}, y_{3}) = (x, 6), {(x}_{4}, y_{4}) = (3, 5) .

⇒ 1 + x 2, 2 + 6 2 = 4 + 3 2, y + 5 2

⇒ 1 + x 2, 4 = 72, y + 5 2

Equating the coordinates from both sides,

1 + x 2 = 72

⇒

1+x=7

⇒

x = 6

Similarly, for y -coordinates,

y + 5 2 = 4

⇒

y+5=8

\Rightarrow

y=3
Hence, the required values of x and y are 6, and 3 respectively.
Therefore, the correct answer is option (3).

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