If coordinates of two adjacent vertices of a parallelogram are (3, 2) and (1, 0) and diagonals bisect each other at (-2, 5), find coordinates of the other two vertices?

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(-7, 8) and (-5, 10)

(7, -8) and (-5, -10)

(7, 8) and (5, 10)

(-7, 8) and (5, 10)

answer is A.

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Detailed Solution

We have given two adjacent vertices of a parallelogram (3, 2) and (1, 0) let us name these vertices as A and B. Also, we have given that the diagonals bisect each other at (-2, 5).
In the below figure, we have drawn a parallelogram ABCD with diagonals AC and BD bisecting each other at point E.
Question Image

Let us assume the coordinates of point C and D as:
C (xC,yC) & D (xD,yD)
And marking these coordinates in the above diagram of parallelogram ABCD we get,
Question Image

Now, we are going to add the x coordinates of vertices A and C and then divide this addition by 2.

\frac{3 + x_{c}}{2}

As diagonals bisect each other so E is the midpoint of AC and BD. Hence, equating the above to -2 we get,

\frac{3 + x_{c}}{2}

-

2
⇒ 3 + xC = −4
⇒ xC = −4−3 = −7
Adding the y coordinates of vertices, A and C followed by division with 2 we get,

\frac{2 + y_{c}}{2}

Equating the above to 5 we get,

\frac{2 + y_{c}}{2}

= 5
⇒ 2 + yC = 2(5) = 10
⇒ yC = 10 – 2 = 8
Hence, we have got the coordinates of vertex C as (-7, 8).
Now, we are going to find the coordinates of vertex D as follows:
Adding the x coordinates of vertices B and D and then divide this addition by 2.

\frac{1 + x_{D}}{2}

Equating the above to -2 we get,

\frac{1 + x_{D}}{2}

=

-

2
1 + xD = 2(−2) = −4
⇒ xD = −4 −1 = −5
Adding the y coordinates of vertices B and D followed by division with 2 we get,

\frac{0 + y_{D}}{2}

Equating the above to 5 we get,

\frac{y_{D}}{2}

= 5

⇒ yD = 2(5) = 10
Hence, we got the coordinates of vertex D as (-5, 10).
Hence, we found the coordinates of the remaining vertices (-7, 8) and (-5, 10).
Hence, option (1) is the correct option.

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