Question

Name the quadrilateral formed, if any, by the following points, and give reasons for your answers:

A (2, –2), В (7, 3), С (11, –1), D (6, –6)

Answer 1

Given: A(2, –2), B(7, 3), C(11, –1), D(6, –6).

Step-wise calculation:

1. Side lengths:

`AB = sqrt((7 - 2)^2 + (3 - (-2))^2` 

= `sqrt(5^2 + 5^2)` 

= `sqrt(50)`

= `5sqrt(2)`

`BC = sqrt((11 - 7)^2 + (-1 - 3)^2)` 

= `sqrt(4^2 + (-4)^2)`

= `sqrt(32)`

= `4sqrt(2)`

`CD = sqrt((6 - 11)^2 + (-6 - (-1))^2)` 

= `sqrt((-5)^2 + (-5)^2)` 

= `sqrt(50)`

= `5sqrt(2)`

`DA = sqrt((2 - 6)^2 + (-2 - (-6))^2)` 

= `sqrt((-4)^2 + 4^2)` 

= `sqrt(32)`

= `4sqrt(2)`

So AB = CD = `5sqrt(2)` and BC = DA = `4sqrt(2)` opposite sides equal.

2. Slopes to test perpendicularity/parallelism:

Slope AB = `(3 - (-2))/(7 - 2)`

= `5/5`

= 1

Slope BC = `(-1 - 3)/(11 - 7)`

= `-4/4`

= –1

Product slope (AB) × Slope (BC) = 1 × (–1) = –1 

⇒ AB ⟂ BC

Also AB vector = (5, 5) and CD vector = (–5, –5)

⇒ AB ∥ CD (and in opposite direction)

BC = (4, –4) and DA = (–4, 4) ⇒ BC ∥ DA

3. Diagonals:

`AC = sqrt((11 - 2)^2 + (-1 - (-2))^2)` 

= `sqrt(9^2 + 1^2)` 

= `sqrt(82)`

`BD = sqrt((6 - 7)^2 + (-6 - 3)^2)` 

= `sqrt((-1)^2 + (-9)^2)` 

= `sqrt(82)`

Diagonals are equal. 

Opposite sides are equal and parallel, adjacent sides are perpendicular, and diagonals are equal.

Therefore, ABCD is a rectangle.