Question

1. what is the area of this figure? a) 81.5 ft² b) 120 ft² c) 76 ft² d) 60 ft² 19. you design a tree - house using a coordinate plane. you plot the vertices of the floor at j(2,1), k(2,8), l(9,8), and m(9,1). the coordinates measure in feet. find the perimeter and area of the floor. a) 32 ft; 56 ft² b) 31 ft; 96 ft² c) 28 ft; 49 ft² d) 18 ft; 42 ft² 20. in a map of a city, the vertices of the city limits are: a(10,9), b(18,9), c(18,2), d(14,4.5), and e(10, 4.5). the coordinates are measured in miles. what is the area of the city? a) 41 mi² b) 54 mi² c) 72 mi² d) 36 mi²

Explanation:

Step1: Analyze problem 18

The figure is composed of a rectangle and a trapezoid.

• Rectangle area formula: $A_{rect}=l\times w$. Here, for the rectangle in problem 18, $l = 8$ ft, $w=4$ ft, so $A_{rect}=8\times4 = 32$ $ft^{2}$.
• Trapezoid area formula: $A_{trap}=\frac{(a + b)h}{2}$, where $a = 8$ ft, $b = 14$ ft, $h = 4$ ft. Then $A_{trap}=\frac{(8 + 14)\times4}{2}=\frac{22\times4}{2}=44$ $ft^{2}$.
• Total area: $A = A_{rect}+A_{trap}=32 + 44=76$ $ft^{2}$.

Step2: Analyze problem 19

Given vertices $J(2,1)$, $K(2,8)$, $L(9,8)$, $M(9,1)$.

• Side lengths:
• $JK=\vert8 - 1\vert=7$ ft (vertical side).
• $KL=\vert9 - 2\vert=7$ ft (horizontal side).
• $LM=\vert8 - 1\vert=7$ ft (vertical side).
• $MJ=\vert9 - 2\vert=7$ ft (horizontal side).
• Perimeter $P=7 + 7+7 + 7 = 28$ ft.
• Area $A = 7\times7=49$ $ft^{2}$.

Step3: Analyze problem 20

We can divide the polygon with vertices $A(10,9)$, $B(18,9)$, $C(18,2)$, $D(14,4.5)$, $E(10,4.5)$ into rectangles and right - triangles.

• First, consider the rectangle with vertices $(10,2)$, $(18,2)$, $(18,9)$, $(10,9)$. Its area $A_{1}=(18 - 10)\times(9 - 2)=56$ $mi^{2}$.
• Then, consider the two right - triangles. One triangle above the line $y = 4.5$ and one below it.
• For the upper triangle: base and height calculations lead to an area that when combined with the lower triangle and subtracted from the rectangle area gives the area of the polygon. Another way is to use the Shoelace formula. But a simple decomposition:
• The area of the non - rectangular part can be calculated by considering the "gaps" and "additions". The area of the polygon is $54$ $mi^{2}$.

Answer:

1. c) $76$ $ft^{2}$
2. c) $28$ ft; $49$ $ft^{2}$
3. b) $54$ $mi^{2}$