Question

If $\left( {1{\text{ }},2} \right),\left( {4,y} \right),\left( {x,6} \right)$ and $(3,5)$ are the vertices of a parallelogram taken in order , find x and y

Answer 1

Hint : Use the property that diagonals of a parallelogram bisect each other. Then use the concept of midpoint in the coordinates of the centre of both the diagonals and equate the coordinates for the required values.

Complete step-by-step answer :
Lets say A,B,C,D are the vertices of a parallelogram ABCD and O is the midpoint where diagonals AC and BD intersect each other.
$A\left( {1,2} \right)\;B\left( {4,y} \right)\;C\left( {x,6} \right)\;D\left( {3,5} \right)$

Now we have to find the coordinates of midpoint i.e. O
x- coordinate of O = $\dfrac{{{x_1} + {x_2}}}{2}$
y- coordinate of O = $\dfrac{{{y_1} + {y_2}}}{2}$
where $x_1$ = 1 and $x_2$ =x
$y_1$ =2 and $y_2$ =6
we will put values to find the coordinates of point O
x – coordinate of O = $y = 3$ $\dfrac{{1 + x}}{2}$
y – coordinate of O = $\dfrac{{2 + 6}}{2}$ = $\dfrac{8}{2}$ = 4
coordinates of point O =( $\dfrac{{1 + X}}{2}$ , 4 ) ... ....(1)
We will repeat the same process but now we will consider diagonal BD and find the O coordinated by taking B and D coordinates into consideration.
x- coordinate of O = $\dfrac{{{x_1} + {x_2}}}{2}$
y- coordinate of O = $\dfrac{{{y_1} + {y_2}}}{2}$
where $x_1$ = 4 and $x_2$ =3
$y_1$ =y and $y_2$ =5
we will put values to find the coordinates of point O
x – coordinate of O = $\dfrac{{4 + 3}}{2}$ = $\dfrac{7}{2}$
y – coordinate of O = $\dfrac{{y + 5}}{2}$
coordinates of point O = $(\dfrac{7}{2},\dfrac{{y + 5}}{2})$ .....(2)
now we will compare equation 1 and 2
$(\dfrac{{1 + x}}{2},4)$ = $(\dfrac{7}{2},\dfrac{{y + 5}}{2})$
We will compare x and y coordinates and find the value of x and y
$\Rightarrow \dfrac{{1 + x}}{2} = \dfrac{7}{2}$
$\Rightarrow 1 + x = 7$
$\Rightarrow x = 7 - 1$
$\Rightarrow x = 6$
Similarly we will find the value of y
$\Rightarrow 4 = \dfrac{{y + 5}}{2}$
$\Rightarrow 8 = y + 5$
$\Rightarrow y = 8 - 5$
$\Rightarrow y = 3$
Therefore the $x=6$ and $y=3$ is the required answer.

Note : Revise the property of parallelogram that both the diagonals of parallelogram bisect each other. Revise the concept of midpoint of a line with the help of coordinates of endpoints of the line. Also mark the vertices ethier in clockwise direction or counterclockwise directions