Question

Plot the points A(2, 1), B(2, –4), C(–3, –4) and D(–3, 1). What kind of quadrilateral is ABCD? Also find its area.

Answer 1

Given: A(2, 1), B(2, –4), C(–3, –4), D(–3, 1).

Step-wise calculation:

1. Compute side vectors:

AB = B – A

= 2 – 2, –4 – 1

= 0, –5

BC = C – B

= –3 – 2, –4 – (–4)

= –5, 0 

CD = D – C

= –3 – (–3), 1 – (–4)

= 0, 5 

DA = A – D

= 2 – (–3), 1 – 1 

= 5, 0

2. Compute side lengths:

|AB| = `sqrt(0^2 + (-5)^2)`

= 5 

|BC| = `sqrt((-5)^2 + 0^2)`

= 5 

|CD| = 5

|DA| = 5

All four sides are equal.

3. Check perpendicularity (adjacent sides):

AB · BC = (0)(–5) + (–5)(0) 

= 0

AB ⟂ BC   ...(Right angle at B) 

Thus, adjacent sides are perpendicular, so the figure has equal sides and right angles.

4. Diagonals (optional check):

AC = C – A

= –5, –5

|AC| = `sqrt(50)`

BD = D – B

= –5, 5

|BD| = `sqrt(50)`

AC · BD = (–5)(–5) + (–5)(5)

= 0 

Diagonals are equal in length and perpendicular.

5. Area: Side length = 5.

So, Area = side² 

= 5² 

= 25 square units. 

Equivalently area = `(d_1 · d_2)/2`

= `(sqrt(50) · sqrt(50))/2` 

= `50/2`

= 25

ABCD is a square.

Area (ABCD) = 25 square units.