Question

If A (1,2), B (4,y), c (x,6), D (3,5) are the vertices of
a parallelogram. Find the value of x and y
​

Answer 1

✰ The value of x is 7 and y is -2✰

Step-by-step explanation:

Given vertices: A(1,2),B(4,y),C(x,6) and D(3,5) are the vertices of a parallelogram.

To find: Value of x and y

Required Formula:

• Co ordinates of mid points = (x₁ + x₂/2, y₁ + y₂/2)

Solution:

Co-ordinate of mid points AC = (x₁ + x₂/2, y₁ + y₂/2)

Co-ordinate of mid points BD = (x₁ + x₂/2, y₁ + y₂/2)

❍ Let A(x₁,y₁) = (1,2),C(x₂,y₂) = (x,6) and B(x₁,y₁) = (4,y) ,D(x₂,y₂) = (3,5)

Putting the values,

Finding the value of x,

$\\ \\ :\implies\sf{AC =\bigg(\frac{x_1 + x_2}{2},\frac{y_1+y_2}{2} \bigg)}\\ \\ :\implies\sf{AC = \bigg( \frac{1 + x}{2}, \frac{2 + 6}{2} \bigg)} \\ \\ :\implies\sf{AC = \bigg( \frac{1 + x}{2} , \frac{ \cancel{8}}{ \cancel{2}} \bigg)} \\ \\ :\implies\sf{AC = \bigg( \frac{1 + x}{2} ,4 \bigg) } \\ \\ :\implies\sf{ \frac{1 + x}{2} =4} \\ \\ :\implies\sf{1 + x = 4 \times 2} \\ \\ :\implies\sf{1 + x = 8} \\ \\ :\implies\sf{x = 8 - 1} \\ \\ :\implies \underline{ \boxed{\frak{x = 7}}} \: \pink{ \bigstar}$

Now,finding the value of y.

$\\ \\ :\implies\sf{ BD=\bigg(\frac{x_1 + x_2}{2},\frac{y_1+y_2}{2} \bigg)}\\ \\ :\implies\sf{ BD=\bigg( \frac{4 + 3}{2} , \frac{ y + 5 }{2} \bigg) } \\ \\ :\implies\sf{ BD=\bigg( \frac{7}{2} , \frac{y + 5}{2} \bigg)} \\ \\ :\implies\sf{ \frac{7}{2} = \frac{y + 5}{2} } \\ \\ :\implies\sf{ \frac{7}{ \cancel{2}}\times \cancel{ 2}=y + 5} \\ \\ :\implies\sf{ 7 =y + 5} \\ \\ :\implies\sf{y=5 - 7} \\ \\ :\implies \underline{ \boxed{\mathfrak{y= - 2}}} \: \blue{ \bigstar}$

Hence,

• The value of x is 7 and y is -2.

Answer 2

Answer

Let A(1,2), B(4,y),C(x,6) and D(3,5) are the vertices of a parallelogram ABCD

.AC and BD are the diagonals .

O is the midpoint of AC and BD.

If O is the mid-point of AC ,then the coordinates of O are =(

2

1+x

,

2

2+6

)=(

2

x+1

,4)

If O is the mid-point of BD then coordinates of O are (

2

4+3

,

2

5+y

)=(

2

7

,

2

5+y

)

Since both coordinates are of the same point O

∴

2

1+x

=

2

7

⇒1+x=7

⇒x=7−1=6

∴

2

5+y

=4

⇒5+y=8

⇒y=8−5=3

Hence x=6 and y=3.