Question

Show that the following points are the vertices of a rectangle:

A(4, 2), В(0, –4), С(–3, –2), D(1, 4)

Answer 1

Given: A(4, 2), B(0, –4), C(–3, –2), D(1, 4).

Step-wise calculation:

1. Compute side vectors:

AB = B – A

= (0 – 4, –4 – 2)

= (–4, –6)

BC = C – B

= (–3 – 0, –2 – (–4))

= (–3, 2)

CD = D – C

= (1 – (–3), 4 – (–2)) 

= (4, 6)

DA = A – D

= (4 – 1, 2 – 4)

= (3, –2)

2. Opposite sides are equal and parallel:

CD = –AB (CD = (4, 6), AB = (–4, –6))

⇒ AB || CD and |AB| = |CD|.

DA = –BC (DA = (3, −2), BC = (–3, 2))

⇒ BC || DA and |BC| = |DA|.

Therefore, ABCD is a parallelogram.

3. Check for a right-angle dot product of adjacent sides:

AB · BC = (–4)(–3) + (–6)(2)

= 12 – 12

= 0

A zero dot product means AB ⟂ BC.

ABCD is a parallelogram with one right angle, hence it is a rectangle.