Question

The fourth vertex D of a parallelogram ABCD whose three consecutive vertices
are A (1,2, B (3,6 ) and C (5,10 ) is
(a) (2,2)
(b) (3,2)
(c) (5,6)
(d) (4,3)​

Answer 1

Step-by-step explanation:

let coordinate x,y

1+5/2=3+x/2

1+5=3+x

3=x

similar you can find y coordinate

but answer would be 3,2

Answer 2

In parallelogram we know that opposite sides are parallel and equal to each other , while parallel lines have equal slope


Therefore slope of bc will be equal to slope of ad and bc will be equal to ad

Now the slope of bc can be calculated by using the formula :

Slope= Change in y/Change in x

Therefore slope of bc will be(10-6)/(5-3)

Which is equal to 4/2



Now the slope of ad will also be 4/2

Now let the coordinates of vertex d be (x,y)


Therefore now slope of ad can be calculated by using formula :slope = change in y/ change in x
This value will be equal to 4/2 (the slope of bc)


Substitute the values

4/2 =( y-2)/(x-1)

4x-4=2y-4

2x=y (equation 1)

Now you got the relation of x and y and now by using the distance formula calculate the distance of bc , it will be equal to the distance of ad ( property of a parallelogram)

By applying distance formula i.e d= d=√((x2-x1)²+(y2-y1)²)

Now substitute the value of x1,x2,y1,y2 respect to line bc

Now d d=√((5-3)²+(10-6)²)

You will get the answer that distance between b,c will be equal to d=√20

Now d=√20 will also be the distance of a,d

Now apply the distance formula with respect to line ad ,

d=√((x-1)²+(y-2)²)

Which will be equal to d=√20
And we already know the relation between x and y ( marked as equation 1)

i.e 2x=y

Substitute this value and square on both the side

d^2=(x-1)²+(2x-2)²

And we know that the distance of this line will be equal to d=√20
[ad=bc(property of parallelogram)]

Therefore now 20= x^2+1 -2x +4x^2+4-8x

20= 5x^2 -10x +5

( divide by 5 on both the sides )

4=x^2-2x +1

0=x^2-2x -3

Now apply the quadratic equation x=(-b+-√b^2-4ac) / 2a.

x= (-(-2) +- √(-2)^2 - 4(1)(-3) ) / 2

You will get 2 possible answers for x , x=-1 , 3

Now if x=-1 then by equation 1 , y will be -2

But if x=3 , y will be 6

Now we have 2 solutions for coordinate of d i.e (-1,-2) and (3,6)

Now in the question itself it is given that (3,6) are the coordinates for vertex b

So the solution d=(3,6) is not possible

Therefore the coordinates of point d must be (-1,-2) to make a parallelogram


Please mark this answer as brainiest answer if you understood...



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