Question

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

A(1, –1), В(–2, 2), C(4, 8), D(7, 5)

Answer 1

Given: A(1, –1), B(–2, 2), C(4, 8), D(7, 5).

Step-wise calculation:

1. Compute the side vectors:

AB = B – A

= (–2 – 1, 2 – (–1))

= (–3, 3)

BC = C – B

= (4 – (–2), 8 – 2)

= (6, 6)

CD = D – C

= (7 – 4, 5 – 8)

= (3, –3)

DA = A – D

= (1 – 7, –1 – 5)

= (–6, –6)

2. Opposite sides are equal and parallel:

CD = –AB since (3, –3) = –1·(–3, 3)).

So, AB || CD and |AB| = |CD|.

DA = –BC since (–6, –6) = –1·(6, 6)).

So, BC || DA and |BC| = |DA|.

Hence, ABCD is a parallelogram.

3. Show one angle is a right angle dot product = 0:

AB · BC = (–3)(6) + (3)(6)

= –18 + 18

= 0

Therefore, AB ⟂ BC, so angle ABC = 90°.

Diagonals AC and BD:

AC = (3, 9)

BD = (9, 3)

|AC|² = 90 = |BD|²

So, diagonals are equal another check that the parallelogram is a rectangle.

Since opposite sides are equal and parallel (parallelogram) and one interior angle is 90°, ABCD is a rectangle.