Question

16. Show that the opposite sides of a quadrilateral with vertices A(-2 ,-4),
B(5 , -1), C(6 , 4) and D(-1, 1) taken in order are parallel.

Answer 1

two lines are said to parallel when slope of these are same.
I mean,if L₁ and L₂ are two parallel lines then, slope of line L₁ = slope of line L₂.

so, we have to find out slope of AB and CD also slop of line AD and BC .
slope of line AB = (-1 + 4)/(5 + 2) = 3/7
slope of line CD = (1 - 4)/(-1 - 6) = -3/-7 = 3/7
Here, slope of line AB = slope of line CD = 3/7
∴ AB is parallel to BC .

Similarly, slope of line AD = (1 +4)/(-1 + 2) = 5/1 = 5
slope of line BC = (4 + 1)/(6 - 5) = 5/1 = 5
Here, slope of line AD = slope of line BC = 5
∴ AD is parallel to BC

Answer 2

Given ABCD is a Quadrilateral .

We have to show that

Diagonals AC and BD bisect each

other .

i ) midpoint of A(-2,-4)= (x1,y1)

and C(6,4) = (x2, y2 ) is (x,y)

coordinates of midpoint

= [ (x1+x2)/2 , (y1+y2)/2 ]

= [ (-2+6)/2 , (-4+4)/2]

= ( 4/2, 0 )

= ( 2 , 0 ) -----( 1 )

ii ) midpoint of B(5,-1) and D(-1,1),

coordinates of midpoint

= [ (5-1)/2 , (-1+1)/2 ]

= ( 4/2 , 0 )

= ( 2 , 0 ) ----( 2 )

( 1 ) = ( 2 )

Therefore ,

diagonals bisects each other.

Therefore ,

ABCD is a parallelogram.

Opposite sides are parallel .

AB//DC,

AD //BC

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