If (3, -4) and (-6,5) are the extremities of the diagonal of a parallelogram and (-2,1) is its third vertex then its fourth vertex is
A. (-1,0)
B. (0, -1)
C. (-1,1)
D. None of these
Answer
636.9k+ views
Hint: In this question, coordinates of three vertices of parallelogram are given, let the fourth coordinate be $(x,y)$. Calculate the coordinate for midpoint of diagonals from the known coordinate and equate it with unknown variables.
Complete step-by-step answer:
Let A= (3, -4); C= (-6,5) as coordinate of diagonal AC
Let coordinate of D be $(x,y)$
Since diagonals of parallelogram bisect each other
Also, midpoint of $
({x_1},{y_1})({x_2},{y_2}) = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right) \\
\\
$
Now, midpoint of AC = midpoint of BD
$
\left( {\dfrac{{3 - 6}}{2},\dfrac{{ - 4 + 5}}{2}} \right) = \left( {\dfrac{{ - 2 + x}}{2},\dfrac{{1 + y}}{2}} \right) \\
\left( {\dfrac{{ - 3}}{2},\dfrac{1}{2}} \right) = \left( {\dfrac{{ - 2 + x}}{2},\dfrac{{1 + y}}{2}} \right) \\
{\text{Equating both sides}} \\
$
$
- 2 + x = - 3{\text{ and }}1 + y = 1 \\
x = - 1{\text{ and }}y = 0 \\
$
Coordinates of fourth vertex is (-1,0)
Hence the correct answer is option A.
Note: For these types of questions, where we need to find the coordinates of any point on a parallelogram where 3 coordinates are already known, the best option is always to go through the midpoint approach, that is why it is always suggested to have a good knowledge of parallelograms and its diagonals. Also we can cross check this answer by comparing the slopes of the opposite lines, that is they should be the same.
Complete step-by-step answer:
Let A= (3, -4); C= (-6,5) as coordinate of diagonal AC
Let coordinate of D be $(x,y)$
Since diagonals of parallelogram bisect each other
Also, midpoint of $
({x_1},{y_1})({x_2},{y_2}) = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right) \\
\\
$
Now, midpoint of AC = midpoint of BD
$
\left( {\dfrac{{3 - 6}}{2},\dfrac{{ - 4 + 5}}{2}} \right) = \left( {\dfrac{{ - 2 + x}}{2},\dfrac{{1 + y}}{2}} \right) \\
\left( {\dfrac{{ - 3}}{2},\dfrac{1}{2}} \right) = \left( {\dfrac{{ - 2 + x}}{2},\dfrac{{1 + y}}{2}} \right) \\
{\text{Equating both sides}} \\
$
$
- 2 + x = - 3{\text{ and }}1 + y = 1 \\
x = - 1{\text{ and }}y = 0 \\
$
Coordinates of fourth vertex is (-1,0)
Hence the correct answer is option A.
Note: For these types of questions, where we need to find the coordinates of any point on a parallelogram where 3 coordinates are already known, the best option is always to go through the midpoint approach, that is why it is always suggested to have a good knowledge of parallelograms and its diagonals. Also we can cross check this answer by comparing the slopes of the opposite lines, that is they should be the same.
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