Question

Prove that the points (–4, –3), (–3, 2), (2, 3) and (1, –2) are the vertices of a rhombus.

Answer 1

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

To Prove: A, B, C, D are the vertices of a rhombus.

Proof [Step-wise]:

1. Label the points: A(–4, –3), B(–3, 2), C(2, 3), D(1, –2).

2. Compute the vectors for the four sides:

AB = B – A

= (–3 – (–4), 2 – (–3))

= (1, 5)

BC = C – B

= (2 – (–3), 3 – 2)

= (5, 1)

CD = D – C

= (1 – 2, –2 – 3)

= (–1, –5)

DA = A – D

= (–4 – 1, –3 – (–2))

= (–5, –1)

3. Compute squared lengths of the sides to avoid square roots:

|AB|² = 1² + 5² 

= 1 + 25

= 26

|BC|² = 5² + 1²

= 25 + 1

= 26

|CD|² = (–1)² + (–5)²

= 1 + 25

= 26

|DA|² = (–5)² + (−1)²

= 25 + 1

= 26

Thus, |AB| = |BC| = |CD| = |DA| all sides are equal.

4. Show opposite sides are parallel so the quadrilateral is a parallelogram:

AB = (1, 5) and CD = (–1, –5) = –AB, so AB || CD and AB = CD in length.

BC = (5, 1) and DA = (–5, –1) = –BC, so BC || DA and BC = DA in length.

Hence, opposite sides are equal and parallel, so ABCD is a parallelogram.

5. Combine the facts:

ABCD is a parallelogram with opposite sides parallel.

All four sides are equal from step 3.

A parallelogram with all sides equal is, by definition, a rhombus.

Diagonals are perpendicular:

AC = C – A

= (6, 6)

BD = D – B

= (4, –4)

AC · BD = 6·4 + 6·(–4)

= 0

So, diagonals are perpendicular, a property consistent with a rhombus.

Since all four sides are equal and opposite sides are parallel, ABCD is a rhombus.