show that quadrilateral ABCD is a parallelogram if A(4,8) B(5,5) C(2,4) D(1,7)
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Answer:
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Let A (4,8) = (x1, y1); B (5,5) = (x2, y2);
C (2,4) = (x3, y3) and D (1,7) = (x4, y4)
Distance between two points P (x1, y1) and Q (x2, y2) =
y2-y1x2-x1
The slope of the line AB=
y2-y1x2-x1 [Distance formula]
=
5-85-4
=
-31=-3 .........(i)
The slope of the line DC =
y4-y3x4-x3 = [Distance formula]
=
7-41-2
=
3-1=-3 ..........(ii)
The slope of the line AD=
y4-y1x2-x1 = [Distance formula]
=
7-41-4
=
-1-3=13 ............(iii)
The slope of the line BC=
y3-y2x3-x2 = [Distance formula]
=
4-52-5=-1-3=13
The slope of line AB = The slope od’s the line DC [From (1) and (2)]
∴ AB || DC
The slope of line AD = The slope of the line BC [From(3) and (4)]
∴ AD || BC
Hence, ABCD is a parallelogram.