Question

1. what is the area of this polygon on a coordinate plane?16 square units12 square units8 square units4 square units

Explanation:

Step1: List polygon vertices

Vertices (from graph): $(1,0)$ is a distractor; actual polygon vertices are $(1,4)$, $(1,6)$, $(2,8)$, $(5,6)$, $(5,4)$, $(2,2)$

Step2: Apply shoelace formula

Label points: $(x_1,y_1)=(1,4)$, $(x_2,y_2)=(1,6)$, $(x_3,y_3)=(2,8)$, $(x_4,y_4)=(5,6)$, $(x_5,y_5)=(5,4)$, $(x_6,y_6)=(2,2)$, $(x_7,y_7)=(1,4)$
Compute sum1: $x_1y_2 + x_2y_3 + x_3y_4 + x_4y_5 + x_5y_6 + x_6y_7$
$=(1×6)+(1×8)+(2×6)+(5×4)+(5×2)+(2×4)=6+8+12+20+10+8=64$
Compute sum2: $y_1x_2 + y_2x_3 + y_3x_4 + y_4x_5 + y_5x_6 + y_6x_7$
$=(4×1)+(6×2)+(8×5)+(6×5)+(4×2)+(2×1)=4+12+40+30+8+2=96$

Step3: Calculate area

Area = $\frac{1}{2}|sum1 - sum2| = \frac{1}{2}|64 - 96| = \frac{1}{2}×32=16$

Answer:

16 square units