Question

What is the area of the polygon formed by the ordered pairs (1,1), (2,1), (1,-4), (2,-4)?

Answer 1

5 square units

Explanation:

See the graph

#A(1,1) B(2,1) C(1,-4) D(2,-4)#
So the polygon is a paralelogram whose sides are 1 and 5
Thus the area is `A=b·h` = `5·1 = 5`

Answer 2

Area of parallelogram ABCD is

`A_P = b * a sin theta = color(blue)( 5)1` sq units

Explanation:

Sides A(1,1), B(2,1), C(1,-4), D(2,-4)

`vec(AB) = sqrt((2-1)^2 + (1-1)^2 = 1`

`vec(BC) = sqrt((2-1)^2 + (1+ 4)^2) = sqrt26`

`vec(CD) = sqrt((2-1)^2 + (-4+4)^2)) = 1`

`vec(AD) = sqrt((1-2)^2 + (1 + 4)^2) = sqrt26`

Since opposite sides are equal, it can be a rectangle or a parallelogram.

By finding the slopes of the equal sides, we can confirm the shape.

Slope `m_(AB) = (1-1) / (2-1) = 0`

Slope `m_(BC) = (1+4) / (2-1) = 5`

Since the `m_(AB) * m_(BC) != -1`, it is not a rectangle and it is only a parallelogram.

To find angle `theta`

`tan theta = m_(BC) = 5`

`theta = tan^(-1) 5 = 1.3734^c or 78.69^0`

Area of parallelogram ABCD is

`A_P = b * a sin theta = 1 * sqrt26 * sin(1.3734) =color(blue)( 5` sq units