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Show that Abcd is a Parallelogram If a = (4, 8), B = (5, 5), C = (2, 4), D = (1, 7) - Geometry

Sum

Show that ABCD is a parallelogram if A = (4, 8), B = (5, 5), C = (2, 4), D = (1, 7)

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Solution

Given: A = (4, 8), B = (5, 5), C = (2, 4), D = (1, 7)

To Prove:  AD || BC

AB || DC

Proof:

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) = `("y"_2-"y"_1)/("x"_2-"x"_1)`

The slope of the line AB= `("y"_2-"y"_1)/("x"_2-"x"_1)`  [Distance formula]

=`(5-8)/(5-4)`

=`-3/1=-3`  .........(i)

The slope of the line DC = `("y"_4-"y"_3)/("x"_4-"x"_3)` =  [Distance formula]

=`(7-4)/(1-2)`

=`3/(-1)=-3`  ..........(ii)

The slope of the line AD=`("y"_4-"y"_1)/("x"_2-"x"_1)` =  [Distance formula]

= `(7-4)/(1-4)`

= `(-1)/-3 = 1/3`    ............(iii)

The slope of the line BC=`("y"_3-"y"_2)/("x"_3-"x"_2)` =  [Distance formula]

=`(4-5)/(2-5)=(-1)/(-3) =1/3` 

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.

Concept: Converse: If a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cyclic.
  Is there an error in this question or solution?
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