use the image below for the next two questions.
5 fill in the blank 1 point find the perimeter of the figure. write your answer in standard form. note: type your answer as is. ex. 2x^3 - x^2 + x - 7 do not use any spaces in your answer.
6 fill in the blank 1 point find the area of the figure. write your answer in standard form. note: type your answer as is. ex. 2x^3 - x^2 + x - 7 do not use any spaces in your answer.

Question

Accepted Answer

### Question 5: Perimeter Calculation
# Explanation:
## Step1: Identify all sides
The figure has two sides of length $3x$ (top and right vertical), a side of $3x + 2$ (right vertical extension), a side of $x$ (bottom of the extension), and we need to find the remaining sides by symmetry. The left side of the extension is $3x + 2$, the bottom of the top rectangle: the total bottom length of the top rectangle should be equal to the top length, which is $3x$, but the extension is $x$ wide, so the bottom of the top rectangle (excluding the extension) is $3x - x=2x$? Wait, no, let's re - examine.

Wait, the figure is a composite of a square (or rectangle) on top and a rectangle below. The top rectangle has length $3x$ and width $3x$? No, looking at the diagram: the top part is a rectangle with length $3x$ (horizontal) and height $3x$ (vertical)? Wait, no, the right side of the top part is $3x$, the top is $3x$, then the vertical side on the right of the lower rectangle is $3x + 2$, the lower rectangle has width $x$ and height $3x+2$? Wait, maybe a better way: to find the perimeter, we sum all the outer sides.

Let's list all the outer sides:

- Top: $3x$
- Right - top vertical: $3x$
- Right - middle vertical: $3x + 2$
- Right - bottom vertical: Wait, no, the lower rectangle has a right side of $3x + 2$, a bottom side of $x$, a left side of $3x + 2$, a top side of $x$ (but the top side of the lower rectangle is internal? No, no, perimeter is the outer boundary.

Wait, maybe the correct way: when we have a composite figure, the perimeter is calculated by adding all the outer edges. Let's consider the horizontal and vertical components separately.

Horizontal sides:

- Top: $3x$
- Bottom: The bottom has two parts: the bottom of the top rectangle (which is $3x$) and the bottom of the lower rectangle ($x$)? No, that's not right. Wait, actually, the horizontal sides: the top is $3x$, the bottom of the entire figure: the lower rectangle has a width $x$, and the top rectangle has a width $3x$, but the lower rectangle is attached to the top rectangle, so the total horizontal length at the bottom: the top rectangle's bottom (excluding the part where the lower rectangle is attached) plus the lower rectangle's bottom. Wait, no, let's use the method of "translating" the sides to find the total horizontal and vertical lengths.

For horizontal sides (left - right):

The total length of horizontal sides: the top is $3x$, the bottom: the lower rectangle has a width $x$, and the top rectangle's bottom (the part not covered by the lower rectangle) is $3x - x = 2x$? No, this is getting confusing. Let's do it step by step.

Looking at the diagram again (as per the given labels):

- The top horizontal side: $3x$
- The right - most vertical side: $3x+(3x + 2)=6x + 2$? No, no. Wait, the vertical sides:

On the right: we have two vertical segments: the top part is $3x$ and the lower part is $3x + 2$, so total right vertical length: $3x+(3x + 2)=6x + 2$

On the left: by symmetry, the left vertical length is also $6x + 2$

Horizontal sides:

The top horizontal side: $3x$

The bottom horizontal side: the lower rectangle has a width $x$, and the top rectangle's bottom (the part not covered by the lower rectangle) is $3x - x=2x$? No, that's incorrect. Wait, the lower rectangle is attached to the top rectangle. The top rectangle has a horizontal length of $3x$, and the lower rectangle has a horizontal length of $x$. So the total bottom horizontal length is $3x$ (because the lower rectangle is attached to the top rectangle, so the bottom of the top rectangle and the bottom of the lower rectangle: the bottom of the top rectangle is $3x$, but the lower rectangle is $x$ wide, so the "exposed" bottom of the top rectangle is $3x - x$, and the bottom of the lower rectangle is $x$, so total bottom horizontal length is $(3x - x)+x = 3x$. Wait, that can't be.

Wait, maybe a better approach: the perimeter of a composite figure can be found by adding the perimeters of the individual figures and subtracting twice the length of the common side (since the common side is internal and counted twice). But in this case, the common side between the top rectangle and the lower rectangle is $x$ (the width of the lower rectangle).

The top rectangle: let's assume it's a rectangle with length $3x$ and width $3x$ (no, the right side of the top rectangle is $3x$, the top is $3x$, so it's a square with side $3x$, perimeter of top square: $4\times3x = 12x$, but we have a lower rectangle attached. The lower rectangle has length $x$ and width $3x + 2$, perimeter of lower rectangle: $2(x+(3x + 2))=2(4x + 2)=8x + 4$. But the common side between them is $x$ (the width of the lower rectangle), so we subtract $2x$ (since it's counted in both perimeters).

So total perimeter $=12x+(8x + 4)-2x=18x + 4$? Wait, no, that's not correct.

Wait, let's look at the diagram again. Let's list all the outer edges:

- Top: $3x$
- Right: $3x+(3x + 2)=6x + 2$
- Bottom: $3x$
- Left: $6x + 2$
- Wait, no, the bottom is not $3x$. Wait, the lower rectangle has a bottom edge of $x$, and the top rectangle has a bottom edge of $3x$, but the lower rectangle is attached to the top rectangle, so the total bottom edge is $3x$ (the top rectangle's bottom) plus $x$ (the lower rectangle's bottom)? No, that would be double - counting.

Wait, I think I made a mistake. Let's use the "walk around" method:

Start at the top - left corner, move right: $3x$

Move down: $3x$

Move right: $3x + 2$

Move down: Wait, no, the lower rectangle's height is $3x + 2$? No, the vertical side on the right of the lower rectangle is $3x + 2$, then move left: $x$

Move up: $3x + 2$

Move left: $3x - x=2x$? No, this is getting too confusing. Wait, maybe the correct way is:

The figure's perimeter is calculated as follows:

The horizontal sides:

- Top: $3x$
- Bottom: The bottom has two parts: the bottom of the top rectangle (which is $3x$) and the bottom of the lower rectangle ($x$)? No, that's wrong. Wait, the lower rectangle is attached to the top rectangle, so the bottom of the top rectangle and the top of the lower rectangle are the same line (internal), so we don't count them.

The vertical sides:

- Right: $3x+(3x + 2)=6x + 2$
- Left: $6x + 2$ (by symmetry)

The horizontal sides:

- Top: $3x$
- Bottom: $3x + x=4x$? No, I think I need to re - evaluate.

Wait, maybe the correct perimeter is $2\times(3x)+2\times(3x+(3x + 2)+x)$? No, that's not right.

Wait, let's consider the figure as a combination of a rectangle (top) with length $3x$ and width $3x$ and a rectangle (bottom) with length $x$ and width $3x + 2$. But when they are combined, the side where they are joined (the width of the bottom rectangle, $x$) is internal.

The perimeter of the top rectangle: $2(3x + 3x)=12x$

The perimeter of the bottom rectangle: $2(x+(3x + 2)) = 2(4x + 2)=8x+4$

The total perimeter of the composite figure is the sum of the perimeters of the two rectangles minus $2x$ (because the common side $x$ is counted twice, once in each rectangle's perimeter, so we subtract $2x$ to get the outer perimeter).

So total perimeter $=12x+(8x + 4)-2x=18x + 4$? Wait, no, that gives $18x + 4$, but let's check with another approach.

Wait, looking at the diagram, the horizontal sides:

- Top: $3x$
- Bottom: $3x + x=4x$ (the bottom of the top rectangle is $3x$, the bottom of the lower rectangle is $x$, and since they are attached, we can consider the total bottom length as $3x + x$)

- The two vertical sides on the left and right: each vertical side is $3x+(3x + 2)=6x + 2$, so two vertical sides: $2\times(6x + 2)=12x + 4$

Then total perimeter $=3x + 4x+12x + 4=19x + 4$? No, this is inconsistent.

Wait, maybe the correct way is:

The figure has:

- Horizontal sides: top $3x$, bottom $3x$, and the two horizontal sides of the lower rectangle: $x$ (left and right? No, the lower rectangle is vertical. Wait, I think I misinterpreted the diagram. Let's assume the top part is a square with side $3x$, and the lower part is a rectangle with width $x$ and height $3x + 2$. The perimeter is calculated as follows:

Perimeter $= 3x+3x+(3x + 2)+x+(3x + 2)+3x$

Wait, let's add them:

$3x+3x = 6x$

$6x+(3x + 2)=9x + 2$

$9x + 2+x = 10x + 2$

$10x + 2+(3x + 2)=13x + 4$

$13x + 4+3x=16x + 4$

No, this is also wrong.

Wait, maybe the diagram is a T - shape. The top bar has length $3x$ and height $3x$, and the vertical bar has width $x$ and height $3x + 2$. To find the perimeter, we calculate the perimeter of the top bar plus the perimeter of the vertical bar minus twice the width of the vertical bar (since the overlapping part is counted twice).

Perimeter of top bar: $2(3x + 3x)=12x$

Perimeter of vertical bar: $2(x+(3x + 2))=8x + 4$

Overlap: $2x$ (the width of the vertical bar, counted twice in the sum of perimeters)

So total perimeter $=12x+8x + 4-2x=18x + 4$

Wait, but let's check with the "walk around" method:

Start at the top - left corner:

- Move right: $3x$ (top of top bar)
- Move down: $3x$ (right of top bar)
- Move right: $3x + 2$ (right of vertical bar)
- Move down: Wait, no, the vertical bar's height is $3x + 2$, so from the bottom of the top bar, we move down $3x + 2$ along the right of the vertical bar, then move left $x$ (bottom of vertical bar), then move up $3x + 2$ (left of vertical bar), then move left $3x - x = 2x$ (bottom of top bar, left part), then move up $3x$ (left of top bar).

Now sum these lengths:

$3x$ (top)+$3x$ (right of top)+$3x + 2$ (right of vertical)+$x$ (bottom of vertical)+$3x + 2$ (left of vertical)+$2x$ (bottom of top - left)+$3x$ (left of top)

Now add them:

$3x+3x=6x$

$6x+(3x + 2)=9x + 2$

$9x + 2+x = 10x + 2$

$10x + 2+(3x + 2)=13x + 4$

$13x + 4+2x = 15x + 4$

$15x + 4+3x=18x + 4$

Ah, so the perimeter is $18x + 4$

## Step 2: Re - check
Wait, maybe there's a simpler way. The T - shape: the top rectangle has length $3x$ and width $3x$, the vertical rectangle has length $x$ and width $3x + 2$. The perimeter of the T - shape is equal to the perimeter of the top rectangle plus the perimeter of the vertical rectangle minus $2x$ (the two sides where they are joined).

Perimeter of top rectangle: $2(3x + 3x)=12x$

Perimeter of vertical rectangle: $2(x+(3x + 2)) = 8x+4$

Subtract $2x$ (the overlapping sides): $12x + 8x+4-2x = 18x + 4$

So the perimeter is $18x + 4$

# Answer: $18x + 4$

### Question 6: Area Calculation
# Explanation:
## Step 1: Identify the two rectangles
The figure is composed of two rectangles: the top rectangle and the bottom rectangle.

The top rectangle: length $l_1 = 3x$ and width $w_1=3x$ (assuming the top part is a rectangle with length $3x$ and width $3x$)

The bottom rectangle: length $l_2=x$ and width $w_2 = 3x + 2$

## Step 2: Calculate the area of each rectangle
The area of a rectangle is given by $A = l\times w$

Area of top rectangle $A_1=3x\times3x = 9x^{2}$

Area of bottom rectangle $A_2=x\times(3x + 2)=3x^{2}+2x$

## Step 3: Calculate the total area
The total area $A = A_1+A_2$

$A=9x^{2}+3x^{2}+2x=12x^{2}+2x$

# Answer: $12x^{2}+2x$

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QUESTION IMAGE

Question

Explanation:

Question 5: Perimeter Calculation

Step1: Identify all sides

Step 1: Identify the two rectangles

Step 2: Calculate the area of each rectangle

Step 3: Calculate the total area

Answer:

Question 6: Area Calculation