Question

find the area of the figure below.
(there is a coordinate grid with a figure labeled with points p, h, g, r, b, s, t. the x-axis and y-axis are marked with grid lines. the figure appears to be a composite shape. at the bottom, there is a box to fill in the area with the label area = units squared)

Explanation:

Step1: Identify figure vertices

Vertices (coordinates): $P(-4,9)$, $T(2,6)$, $S(5,6)$, $B(5,1)$, $R(3,-5)$, $G(-2,-5)$, $H(-3,1)$

Step2: Split figure into 3 parts

Divide into: Triangle $PTH$, Trapezoid $TSBH$, Trapezoid $HGRB$

Step3: Calculate area of $\triangle PTH$

Base $TH = |2 - (-3)| = 5$, Height $= 9 - 1 = 8$
Area: $\frac{1}{2} \times 5 \times 8 = 20$

Step4: Calculate area of trapezoid $TSBH$

Bases: $TS = |5 - 2| = 3$, $HB = |5 - (-3)| = 8$, Height $= 6 - 1 = 5$
Area: $\frac{1}{2} \times (3 + 8) \times 5 = \frac{1}{2} \times 11 \times 5 = 27.5$

Step5: Calculate area of trapezoid $HGRB$

Bases: $HG = |-2 - (-3)| = 1$, $RB = |5 - 3| = 2$, Height $= 1 - (-5) = 6$
Area: $\frac{1}{2} \times (1 + 2) \times 6 = \frac{1}{2} \times 3 \times 6 = 9$

Step6: Sum all areas

Total Area: $20 + 27.5 + 9$

Answer:

56.5 units squared