Question

unit 1: shape and space /10 name: lesson 1-2 quiz: area and perimeter calculations must include formula, substitution and solution steps for ks. show all work and include proper units for your answers. culate both the area and perimeter of the owing polygon. trapezoid diagram with 12 m, 7 m, 4 m, 8 m, 18 m a=(b1+b2)h/2 (handwritten) e area of the shaded region in the figure. composite figure diagram with 5 cm, 3 cm, 2 cm, 6 cm

Explanation:

First Problem: Area and Perimeter of Trapezoid

Step 1: Identify the formula for the area of a trapezoid

The formula for the area of a trapezoid is \( A=\frac{(b_1 + b_2)h}{2} \), where \( b_1 \) and \( b_2 \) are the lengths of the two parallel sides (bases) and \( h \) is the height (the perpendicular distance between the bases). For the given trapezoid, \( b_1 = 12\space m \), \( b_2=18\space m \), and \( h = 4\space m \).

Substitute the values into the formula:
\( A=\frac{(12 + 18)\times4}{2} \)

Step 2: Calculate the area

First, add the bases: \( 12+ 18=30 \)
Then multiply by the height: \( 30\times4 = 120 \)
Then divide by 2: \( \frac{120}{2}=60 \)
So the area of the trapezoid is \( 60\space m^2 \)

Step 3: Calculate the perimeter of the trapezoid

The perimeter of a polygon is the sum of the lengths of all its sides. The sides of the trapezoid are \( 12\space m \), \( 7\space m \), \( 18\space m \), and \( 8\space m \)
Perimeter \( P=12 + 7+18 + 8 \)
Add the numbers: \( 12+7 = 19 \), \( 19+18=37 \), \( 37 + 8=45 \)
So the perimeter is \( 45\space m \)

Second Problem: Area of the Shaded Region (Composite Figure)

We can divide the composite figure into two rectangles. Let's call the first rectangle (the lower one) with length \( l_1=(5 + 2)\space cm=7\space cm \) and width \( w_1 = 3\space cm \), and the second rectangle (the upper one) with length \( l_2=2\space cm \) and width \( w_2=(6 - 3)\space cm = 3\space cm \)

The formula for the area of a rectangle is \( A = l\times w \)

Step 1: Calculate the area of the first rectangle

\( A_1=l_1\times w_1=7\times3 = 21\space cm^2 \)

Step 2: Calculate the area of the second rectangle

\( A_2=l_2\times w_2=2\times3=6\space cm^2 \)

Step 3: Calculate the total area of the shaded region

The total area \( A = A_1+A_2 \)
\( A=21 + 6=27\space cm^2 \)

Answer:

s:

• Area of Trapezoid: \( 60\space m^2 \), Perimeter of Trapezoid: \( 45\space m \)
• Area of Shaded Region: \( 27\space cm^2 \)