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

#### 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) = (x_{1}, y_{1}); B (5,5) = (x_{2}, y_{2});

C (2,4) = (x_{3}, y_{3}) and D (1,7) = (x_{4}, y_{4})

Distance between two points P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) = `("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.