Question

The three vertices of a parallelogram ABCD are A(–3, –4), B(2, –2) and C(2, 6). Plot these points on a graph paper and find the co-ordinates of the fourth vertex. Also find the area of the parallelogram.

Answer 1

The coordinates of the fourth vertex are D(–3, 4) and the area of the parallelogram is 40 square units.

1. Find the fourth vertex D(x, y)

In a parallelogram, the diagonals bisect each other. This means the midpoint of diagonal AC is the same as the midpoint of diagonal BD.

Midpoint of AC:

`M = ((-3 + 2)/2, (-4 + 6)/2)`

= `(- 1/2, 1)`

Midpoint of BD:

Using vertex B(2, –2) and unknown D(x, y):

`((2 + x)/2, (-2 + y)/2)`

= `(- 1/2, 1)`

Equating the coordinates:

1. `(2 + x)/2 = -1/2`

⇒ 2 + x = –1

⇒ x = –3

2. `(-2 + y)/2 = 1`

⇒ –2 + y = 2

⇒ y = 4

Thus, the coordinates of D are (–3, 4).

2. Calculate the area

The area of a parallelogram can be calculated using the formula:

Area = Base × Height

Base (BC): The segment BC is a vertical line because both x-coordinates are 2.

Length of BC

= |y_C – y_B|

= |6 – (–2)|

= 8 units

Height (h): The height is the perpendicular distance between the vertical lines x = 2 (side BC) and x = –3 (side AD).

h = |x_BC – x_AD|

= |2 – (–3)|

= 5 units

Area = 8 × 5

= 40 sq units

The coordinates of the fourth vertex are D(–3, 4) and the area of the parallelogram is 40 square units.