Question

Image a coordinate plane with points plotted at ordered pairs Q(-4,4) R(2,4) S(5,-3) T(2,-3) and U (-4,-3) what is the area of the polygon QRSTU

Answer 1

In counterclockwise order that's

R(2,4), Q(-4,4), U(-4,3), T(2,-3), S(5,-3)

That's a trapezoid, height 7 upper base 6, lower base 9, so average base 15/2 and area 7(15)/2 = 105/2 = 52.5.

Answer: 52.5

The shoelace theorem gives the answer without thinking. We write the list on top of itself shifted and add up the cross produces:

R(2,4), Q(-4,4), U(-4,-3), T(2,-3), S(5,-3)

Q(-4,4), U(-4,-3), T(2,-3), S(5,-3), R(2,4),

$A = \frac 1 2 | 2(4)-(4)(-4) + -4(-3)-(-4)(4) + -4(-3)- -3(2) + 2(-3)-(-3)(5) + 5(4)- -3(2)| = \frac{105}{2} \quad\checkmark$