Question

if A(-2,1), B(9,0) , C(4,b), D(1,a) are the vertices of a parallelogram ABCD then find the value of a and b

Answer 1

We will use the following concept :

1.) parallel lines have equal gradient.

2.) gradient = change in y /change in x

AB // DC and AD // BC

The gradient of AB :

(0 - 1) / (9 + 2) = - 1/11

Gradient of AB = Gradient of DC

Therefore :

(b - a) / (4-1) = - 1/11

(b - a) / 3 = - 1/11

11b - 11a = - 3

11a - 11b = 3................i)

Gradient of AD = gradient of BC therefore :

(0 - b) / (9 - 4) = (1 - a) / (-2-1)

-b / 5 = (1 - a) / - 3

3b = 5 - 5a

3b + 5a = 5.................ii)

Solving for a and b from i and ii simultaneously :

5a + 3b = 5
11a - 11b = 3

Lets eliminate a by multiplying i 5 and ii by 11 then subtracting i from ii

We get :

55a + 33b = 55
-
55a - 55b = 15

88b = 40

b = 40/88 = 5 / 11

Substituting the value of b in ii to get a

5a + 3 × 5/11 = 5

5a = 55/11 - 15/11

5a = 40/11

a = 40 / 11 × 1/5

a = 8/11

Answer :

a = 8/11

b = 5/11