Question

What is the perimeter of a rectangle with vertices A(-1, 1), B(3, 4), C(6, 0), and D(2, -3)?

Answer 1

Perimeter of the square `P = color(green)(20`

Explanation:

Given A (-1,1), B (3,4), C (6,0), D (2,-3)

To find perimeter of the rectangle ABCD.

Perimeter of quadrilateral `P = VEC(AB) + vec(BC) + vec(CD) + vec(DA)`

Using distance formula,

`vec(AB) = sqrt((-1-3)^2 + (4-1)^2) = color(purple)(5`

`vec(BC) = sqrt((6-3) + (0-4)^2) = color (purple)(5`

`vec(CD) = sqrt((6-2)^2 + (0+3)^2) = color (purple)(5`

`vec(DA) = sqrt((2+1)^2 + (-3-1)^2) = color(purple)(5`

Since all the 4 sides are equal, it can be a square or a rhombus.

Slope `m_(AB) = (4-1) / (3+1) = 3/4 = -(1/m_(DA))`

Slope `m_(BC) = (0-4) / (6-3) = -4/3 = -(1/m_(AB))`

Slope `m_(CD) = (-3-0) / (2-6) = 3/4 = (-1/m_(BC))`

Slope `m_(DA) = (-3-1) / (2+1) = -4/3 = -(1/m_(CD))`

Since the sides are at right angle, it is a square.

Perimeter of the square `P = 5 + 5 + 5 + 5 = color(green)(16`