find the area of each figure. (example 1)
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7. find the area of the shaded region.
8. find the area of the shaded region.

Question

Accepted Answer

Let's solve problem 1 first:

# Explanation:
The figure is composed of a trapezoid (or a hexagon? Wait, no, looking at the diagram: it's a rectangle and a trapezoid? Wait, no, the top part is a trapezoid with bases 6 yd and 12 yd, height 6 yd, and the bottom part is a square (or rectangle) with side 12 yd and height 12 yd? Wait, no, the diagram shows: the top figure is a trapezoid (bases 6 yd and 12 yd, height 6 yd) and the bottom is a rectangle (length 12 yd, height 12 yd). Wait, actually, the total figure is a combination of a trapezoid (top) and a rectangle (bottom).

## Step 1: Area of the trapezoid
The formula for the area of a trapezoid is $ A = \frac{(a + b)h}{2} $, where $ a $ and $ b $ are the lengths of the two parallel sides (bases), and $ h $ is the height (the perpendicular distance between the bases).
Here, $ a = 6 $ yd, $ b = 12 $ yd, $ h = 6 $ yd.
So, area of trapezoid $ = \frac{(6 + 12) \times 6}{2} = \frac{18 \times 6}{2} = 54 $ square yards.

## Step 2: Area of the rectangle
The formula for the area of a rectangle is $ A = l \times w $, where $ l $ is the length and $ w $ is the width.
Here, $ l = 12 $ yd, $ w = 12 $ yd.
So, area of rectangle $ = 12 \times 12 = 144 $ square yards.

## Step 3: Total area
Add the area of the trapezoid and the area of the rectangle.
Total area $ = 54 + 144 = 198 $ square yards.

# Answer:
The area of the figure is $ \boxed{198} $ square yards.

Now problem 2:

The figure is a combination of a triangle and a rectangle? Wait, the diagram shows a right triangle? Wait, no, the figure has a right angle at the bottom, with base 4 ft, height 3 ft (the vertical side), and then a slanted side. Wait, maybe it's a trapezoid? Wait, the figure has sides: 4 ft (base), 3 ft (vertical), 5 ft (slant), 6 ft (hypotenuse). Wait, maybe it's a trapezoid with bases 3 ft and 6 ft? No, better to split it into a rectangle and a triangle. Wait, the bottom part is a rectangle with length 4 ft and height 3 ft, and the top part is a triangle with base 4 ft and height (6 - 3) = 3 ft? Wait, no, the area of a trapezoid: the formula for the area of a trapezoid is $ \frac{(a + b)h}{2} $, where $ a $ and $ b $ are the two parallel sides. Wait, the figure has two parallel sides: 3 ft and 6 ft? No, maybe it's a triangle and a rectangle. Wait, the area can also be calculated as the area of a trapezoid with bases 3 ft and 6 ft, and height 4 ft? Wait, no, let's check the coordinates. Wait, the figure is a trapezoid with bases 3 ft (the vertical side) and 6 ft (the slanted side? No, that's not parallel. Wait, maybe it's a right trapezoid? Wait, the right angle is at the bottom, so the two parallel sides are the vertical side (3 ft) and the top side (6 ft)? No, that's not parallel. Wait, maybe the figure is a triangle with base 4 ft and height (3 + 6) = 9 ft? No, that doesn't make sense. Wait, the correct way: the figure is a trapezoid with bases 3 ft and 6 ft, and height 4 ft? Wait, no, let's use the formula for the area of a trapezoid: $ A = \frac{(a + b)h}{2} $, where $ a $ and $ b $ are the lengths of the two parallel sides (the two sides that are parallel), and $ h $ is the distance between them (the height, which is perpendicular to both). Looking at the diagram, the two parallel sides are 3 ft (the vertical side) and 6 ft (the top side)? No, that's not parallel. Wait, maybe it's a triangle and a rectangle. The bottom part is a rectangle with length 4 ft and height 3 ft (area = 4*3 = 12). The top part is a triangle with base 4 ft and height (6 - 3) = 3 ft? Wait, no, the area of the triangle would be (4*3)/2 = 6. Then total area is 12 + 6 = 18? Wait, but let's check with the trapezoid formula. If we consider the two parallel sides as 3 ft and 6 ft, and the height (the horizontal distance) as 4 ft, then area is (3 + 6)*4/2 = 18. Yes, that works. So:

## Step 1: Identify the trapezoid
Bases $ a = 3 $ ft, $ b = 6 $ ft, height $ h = 4 $ ft.

## Step 2: Apply the trapezoid area formula
$ A = \frac{(3 + 6) \times 4}{2} = \frac{9 \times 4}{2} = 18 $ square feet.

# Answer:
The area of the figure is $ \boxed{18} $ square feet.

Problem 3:

The figure is composed of a parallelogram and a trapezoid.

## Step 1: Area of the parallelogram
The formula for the area of a parallelogram is $ A = base \times height $.
Here, base = 7 cm, height = 5 cm (wait, the diagram shows the parallelogram has base 7 cm and height 5 cm? Wait, the parallelogram has sides 10 cm and 5 cm? Wait, no, the parallelogram: base = 7 cm, height = 5 cm? Wait, the diagram: the parallelogram has a base of 7 cm and a height of 5 cm? Wait, no, the parallelogram's area is $ 7 \times 5 = 35 $ cm²? Wait, no, the height is 5 cm, base is 7 cm? Wait, the trapezoid below: bases 7 cm and 14 cm, height 6 cm.

## Step 2: Area of the trapezoid
Formula: $ \frac{(a + b)h}{2} $, where $ a = 7 $ cm, $ b = 14 $ cm, $ h = 6 $ cm.
Area of trapezoid $ = \frac{(7 + 14) \times 6}{2} = \frac{21 \times 6}{2} = 63 $ cm².

## Step 3: Total area
Add the area of the parallelogram and the trapezoid.
Wait, no, the parallelogram: wait, the diagram shows the parallelogram with base 7 cm and height 5 cm? Wait, no, the parallelogram's sides: 10 cm and 5 cm? Wait, maybe the parallelogram's area is $ 10 \times 5 = 50 $ cm²? Wait, I think I made a mistake. Let's re-examine:

The parallelogram: the base is 7 cm, and the height is 5 cm? No, the height is the perpendicular distance. Wait, the diagram shows a right angle between 5 cm and 10 cm? No, the 5 cm is the height. Wait, the parallelogram's area is $ base \times height = 7 \times 5 = 35 $ cm²? Wait, no, the base is 10 cm? Wait, the diagram: the parallelogram has a side of 10 cm and a height of 5 cm, so area is $ 10 \times 5 = 50 $ cm². Then the trapezoid: bases 7 cm and 14 cm, height 6 cm. Area of trapezoid: $ \frac{(7 + 14) \times 6}{2} = 63 $ cm². Then total area: $ 50 + 63 = 113 $ cm²? Wait, no, maybe the parallelogram's base is 7 cm, height 5 cm: $ 7 \times 5 = 35 $, trapezoid: $ \frac{(7 + 14) \times 6}{2} = 63 $, total: $ 35 + 63 = 98 $ cm²? Wait, I'm confused. Let's check the correct approach.

Wait, the figure is composed of a parallelogram (top) and a trapezoid (bottom). The parallelogram has a base of 7 cm and a height of 5 cm (since there's a right angle between 5 cm and 7 cm? No, the 5 cm is the height. So area of parallelogram: $ 7 \times 5 = 35 $ cm². The trapezoid has bases 7 cm and 14 cm, height 6 cm. Area of trapezoid: $ \frac{(7 + 14) \times 6}{2} = \frac{21 \times 6}{2} = 63 $ cm². Total area: $ 35 + 63 = 98 $ cm².

# Explanation:
## Step 1: Area of the parallelogram
The formula for the area of a parallelogram is $ A = base \times height $.
Here, $ base = 7 $ cm, $ height = 5 $ cm.
So, area of parallelogram $ = 7 \times 5 = 35 $ cm².

## Step 2: Area of the trapezoid
The formula for the area of a trapezoid is $ A = \frac{(a + b)h}{2} $, where $ a = 7 $ cm (top base), $ b = 14 $ cm (bottom base), and $ h = 6 $ cm (height).
So, area of trapezoid $ = \frac{(7 + 14) \times 6}{2} = \frac{21 \times 6}{2} = 63 $ cm².

## Step 3: Total area
Add the area of the parallelogram and the area of the trapezoid.
Total area $ = 35 + 63 = 98 $ cm².

# Answer:
The area of the figure is $ \boxed{98} $ square centimeters.

Problem 4:

The figure is a square (or rectangle) with a triangle attached to the bottom.

## Step 1: Area of the square
The square has side 8 ft. Area of square $ = 8 \times 8 = 64 $ square feet.

## Step 2: Area of the triangle
The triangle has base 8 ft (same as the square's side) and height 4 ft. Area of triangle $ = \frac{1}{2} \times base \times height = \frac{1}{2} \times 8 \times 4 = 16 $ square feet.

## Step 3: Total area
Add the area of the square and the area of the triangle.
Total area $ = 64 + 16 = 80 $ square feet.

# Answer:
The area of the figure is $ \boxed{80} $ square feet.

Problem 5:

The figure is a rectangle with a triangle on top.

## Step 1: Area of the rectangle
The rectangle has length 20 m and height 6 m. Area of rectangle $ = 20 \times 6 = 120 $ square meters.

## Step 2: Area of the triangle
The triangle has base $ 20 - 8 - 6 = 6 $ m (since the total length is 20 m, and the two segments are 8 m and 6 m) and height $ 10 - 6 = 4 $ m (since the total height is 10 m, and the rectangle's height is 6 m). Wait, no, the triangle's base is $ 20 - 8 - 6 = 6 $ m, and height is $ 10 - 6 = 4 $ m? Wait, no, the total height is 10 m, so the triangle's height is $ 10 - 6 = 4 $ m. Area of triangle $ = \frac{1}{2} \times base \times height = \frac{1}{2} \times 6 \times 4 = 12 $ square meters. Wait, no, that can't be. Wait, the correct way: the figure is a rectangle (length 20 m, height 6 m) and a triangle (base 20 m, height $ 10 - 6 = 4 $ m)? No, the top part is a triangle with base $ 20 - 8 - 6 = 6 $ m? Wait, the diagram shows: the bottom part is a rectangle with length 20 m, height 6 m. The top part is a triangle with base $ 20 - 8 - 6 = 6 $ m, and height $ 10 - 6 = 4 $ m? No, I think the triangle's base is 20 m, and height is $ 10 - 6 = 4 $ m. Wait, no, the total length is 20 m, so the triangle's base is 20 m, and height is $ 10 - 6 = 4 $ m. Then area of triangle $ = \frac{1}{2} \times 20 \times 4 = 40 $ square meters. Wait, that makes more sense.

## Step 3: Total area
Add the area of the rectangle and the area of the triangle.
Total area $ = 120 + 40 = 160 $ square meters? Wait, no, let's re-examine:

The rectangle: length 20 m, height 6 m: area $ 20 \times 6 = 120 $ m².

The triangle: base $ 20 - 8 - 6 = 6 $ m, height $ 10 - 6 = 4 $ m: area $ \frac{1}{2} \times 6 \times 4 = 12 $ m².

Total area $ = 120 + 12 = 132 $ m²? Wait, I'm confused. Let's check the correct approach.

Wait, the figure is a composite shape: the bottom is a rectangle with length 20 m and height 6 m. The top is a triangle with base $ 20 - 8 - 6 = 6 $ m (since the two horizontal segments are 8 m and 6 m) and height $ 10 - 6 = 4 $ m (since the total height is 10 m, and the rectangle's height is 6 m). So area of triangle is $ \frac{1}{2} \times 6 \times 4 = 12 $ m². Then total area is $ 20 \times 6 + 12 = 120 + 12 = 132 $ m².

# Answer:
The area of the figure is $ \boxed{132} $ square meters.

Problem 6:

The figure is a trapezoid.

## Step 1: Identify the bases and height
The two parallel sides (bases) are $ 1.1 $ cm and $ 5.9 $ cm? Wait, no, the diagram shows: the bottom base is $ 1.1 + 5.9 = 7 $ cm? Wait, no, the bases are $ 3.6 $ cm? No, the formula for the area of a trapezoid is $ \frac{(a + b)h}{2} $, where $ a $ and $ b $ are the two parallel sides (the lengths of the two parallel sides), and $ h $ is the height (the perpendicular distance between them).

Looking at the diagram: the two parallel sides are $ 3.6 $ cm (the vertical side) and... Wait, no, the bases are the two horizontal sides? Wait, the height is $ 4.8 $ cm, and the two bases are $ 1.1 $ cm and $ 5.9 $ cm? Wait, no, the total length of the bottom base is $ 1.1 + 5.9 = 7 $ cm? Wait, no, the diagram shows: the two parallel sides (bases) are $ 3.6 $ cm? No, I think the bases are $ 1.1 $ cm and $ 5.9 $ cm, and the height is $ 4.8 $ cm? Wait, no, the height is the perpendicular distance between the two bases, which are horizontal. Wait, the correct bases are $ 1.1 $ cm and $ 5.9 $ cm, and the height is $ 4.8 $ cm? No, the height is $ 3.6 $ cm? Wait, I'm confused. Let's check

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QUESTION IMAGE

Question

Explanation:

Step 1: Area of the trapezoid

Step 2: Area of the rectangle

Step 3: Total area

Step 1: Area of the parallelogram

Step 2: Area of the trapezoid

Step 3: Total area

Answer:

Step 1: Identify the trapezoid

Step 2: Apply the trapezoid area formula