Question

اد) 8 (ط) (مع
if (-2,-1) (a,0) (4,b) (1,2) are the vertices of a parallelogram then find a and b values
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Answer 1

Answer :

The required values are

a = 1 and b = 3

Given :

• (-2 , -1) , (a , 0) , (4 , b) and (1 , 2) are the vertices of a parallelogram .

To Find :

• The value of a and b

Formula to be used :

If (x₁ , y₁) and (x₂ , y₂) are two points then the coordinates of mid point of the line joining these points is :

$\sf(x,y) = (\dfrac{ x_{1} + x_{2} }{2} ,\dfrac{ y_{1} + y_{2} }{2} )$

Solution :

Let us consider A( -2 , -1) , B(a , 0) , C(4 , b) and D(1 , 2)

Since ABCD is a parallelogram so the diagonals of ABCD bisect each other .

Therefore ,

Mid point of AC = Mid point of BD

$\sf \implies( \dfrac{ - 2 + 4}{2} , \dfrac{ - 1 + b}{2} ) = ( \dfrac{a+ 1}{2} , \dfrac{0 + 2}{2} ) \\ \\ \implies \sf(1,\dfrac{ - 1 + b}{2}) = (\dfrac{a+ 1}{2},1)$

Equating the values of ordered pair we have :

$\sf1 = \frac{a + 1}{2} \\ \implies \sf a + 1 = 2 \\ \implies \sf a = 1$

and

$\sf \implies \dfrac{ - 1 + b}{2} = 1 \\ \sf\implies - 1 + b = 2 \\ \sf \implies b = 3$

Answer 2

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QUESTION :

اد) 8 (ط) (مع

اد) 8 (ط) (معif (-2,-1) (a,0) (4,b) (1,2) are the vertices of a parallelogram then find a and b values..

SOLUTION :

This question can be solved in various methods.

The first method is by using the distance Formula.

But That process is quite lengthy.

So, I am solving this question using a Shortcut method.

Recall the properties of a Parallelogram.

We know that the Diagonals of a Parallelogram bisect each other at a point.

This point is the midpoint of the vertices.

We are going to use that property here.

From the above Question, we can gather the following information.....

Let the vertices of the Parallelogram be M, N, O and P

M => ( -2 , - 1 )

N => ( a , 0 )

O => ( 4 , b )

P => ( 1 , 2 )

M and O are a pair of opposite vertices of the above Parallelogram.

So are N and P

Hence,

Midpoint =>

( -1 , { b - 1 / 2 } ) = ( { a + 1 / 2 } , 1 )

So,

a + 1 = -2

=> a = -3

b - 1 = 2

=> b = 3

So the values of a and b are -3 and 3 respectively.