Question

If A(–2, 1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of its sides.

Answer 1

Answer

In a parallelogram, the diagonals bisect each other.

ABCD is a parallelogram. Diagonals AC and BD bisect each other.

Midpoint of AC = Midpoint of BD

Midpoint formula: (x1 + x2)/2 ; (y1 + y2)/2

(-2+4)/2 ; (1+b)/2 = (a+1)/2 ; (0+2)/2

2/2 ; (1+b)/2 = (a+1)/2 ; 2/2

=> 1 = (a + 1)/2

a = 1

=> (1 + b)/2 = 1

b = 1

So, a = 1 and b = 1.

Using distance formula, you can find the length of the sides of parallelogram.

Distance formula: √[(x2 - x1)² + (y2 - y1)²]

AB = √[(1 + 2)² + (0 - 1)²]

= √10

BC = √[(4 - 1)² + (1 - 0)²]

= √10

As all sides are equal, it is a rhombus.

Answer 2

a = 1

b = 1

length of its sides = 10 units.

All sides are equal, so, it is also a rohmbus.