9. jen wants to paint one wall in her room. there is a door and window in the room. what is the total wall space that needs to be painted?
total wall space:
______
mage.b.2.a: use geometric reasoning and symmetries of regular polygons to develop definitions of rotations, reflections, and translations. identify lines of symmetry and angles of rotations to map a figure onto itself
10. if the figure below follows a reflexive or line of symmetry, then complete the figure.
11. identify the shapes which do not have rotational symmetry from the below figure.
(a) (b) (c)
12. in each of the following figures, a hole was punched in a folded square sheet of paper and then the paper was unfolded. identify the line along which the paper was folded.
a. b. c. d.
figure (d) was created by punching a single hole. how was the paper folded?

Question

9. jen wants to paint one wall in her room. there is a door and window in the room. what is the total wall space that needs to be painted?
total wall space:
______
mage.b.2.a: use geometric reasoning and symmetries of regular polygons to develop definitions of rotations, reflections, and translations. identify lines of symmetry and angles of rotations to map a figure onto itself
10. if the figure below follows a reflexive or line of symmetry, then complete the figure.
11. identify the shapes which do not have rotational symmetry from the below figure.
(a)  (b)  (c)
12. in each of the following figures, a hole was punched in a folded square sheet of paper and then the paper was unfolded. identify the line along which the paper was folded.
a.  b.  c.  d.
figure (d) was created by punching a single hole. how was the paper folded?

Accepted Answer

### Question 9
# Explanation:
## Step1: Find area of the wall
The wall is a rectangle with length $ 7x + 1 $ and height $ 2x + 4 $. The area of a rectangle is $ \text{length} \times \text{height} $, so the area of the wall is $ (7x + 1)(2x + 4) $.
Expanding this: $ (7x)(2x)+(7x)(4)+(1)(2x)+(1)(4)=14x^{2}+28x + 2x+4 = 14x^{2}+30x + 4 $
## Step2: Find area of the door
The door is a rectangle with width $ x $ and height $ x - 1 $. The area of the door is $ x(x - 1)=x^{2}-x $
## Step3: Find area of the window
The window is a rectangle with length $ 2x + 1 $ and height $ x + 1 $. The area of the window is $ (2x + 1)(x + 1) $
Expanding this: $ (2x)(x)+(2x)(1)+(1)(x)+(1)(1)=2x^{2}+2x+x + 1=2x^{2}+3x + 1 $
## Step4: Subtract door and window areas from wall area
The area to be painted is the area of the wall minus the area of the door minus the area of the window.
So, $ (14x^{2}+30x + 4)-(x^{2}-x)-(2x^{2}+3x + 1) $
First, distribute the negative signs: $ 14x^{2}+30x + 4 - x^{2}+x - 2x^{2}-3x - 1 $
Then, combine like terms:
- For $ x^{2} $ terms: $ 14x^{2}-x^{2}-2x^{2}=11x^{2} $
- For $ x $ terms: $ 30x+x - 3x = 28x $
- For constant terms: $ 4 - 1 = 3 $
So the area to be painted is $ 11x^{2}+28x + 3 $
# Answer:
$ 11x^{2}+28x + 3 $

### Question 10
# Explanation:
## Step1: Identify the line of symmetry
The given figure is symmetric about the horizontal line (the base line). To complete the figure, we reflect the existing part (the triangle - like notch and the half - circle) over this horizontal line.
## Step2: Draw the reflected part
The left - most part (the triangle - like notch) will have a mirror image below the horizontal line, and the half - circle is already symmetric about this line (its lower half will be drawn to complete the full - looking shape with the notch's reflection).
# Answer:
The completed figure is the reflection of the given figure over the horizontal line (the base line). The left - hand triangular notch is reflected below the horizontal line, and the semi - circle remains as is (with its lower part implied by the reflection of the notch to form a symmetric figure).

### Question 11
# Explanation:
## Step1: Recall rotational symmetry
A shape has rotational symmetry if it can be rotated by an angle less than $ 360^{\circ} $ and map onto itself.
- For shape (a): A non - equilateral triangle (scalene) has no rotational symmetry because rotating it by any angle less than $ 360^{\circ} $ will not make it look the same.
- For shape (b): A regular hexagon has rotational symmetry. It can be rotated by $ 60^{\circ},120^{\circ},180^{\circ},240^{\circ},300^{\circ} $ and map onto itself.
- For shape (c): The lightning - like shape has no rotational symmetry because rotating it by any angle less than $ 360^{\circ} $ will not make it look the same.
But usually, among these, (a) and (c) are non - rotationally symmetric, but if we consider the most probable (assuming (a) is isoceles or scalene, and (c) is irregular), but from the diagram, (c) and (a) (if (a) is scalene) do not have rotational symmetry. However, typically, the lightning - shaped figure (c) and the non - regular triangle (a) are the ones without rotational symmetry. But if we have to choose, (c) is more obviously non - rotationally symmetric as (a) might be isoceles (but the diagram shows a scalene - like triangle).
## Step2: Conclusion
Shapes (a) and (c) do not have rotational symmetry. But if we consider the options, (a) is a triangle (if scalene) and (c) is the lightning - like shape. So the shapes which do not have rotational symmetry are (a) and (c).
# Answer:
(a) and (c)

### Question 12 (for figure d)
# Explanation:
## Step1: Analyze the holes in figure d
In figure (d), there are four holes, two on the top and two on the bottom, two on the left and two on the right. This suggests that the paper was folded twice, once along the vertical line of symmetry and once along the horizontal line of symmetry (or vice - versa), or along the two diagonals? Wait, no. When we punch a hole in a folded paper, the number of holes is related to the number of layers. If we fold the square paper along both the vertical and horizontal mid - lines (i.e., the paper is folded into four quadrants), punching a hole will result in four holes, one in each quadrant. So the paper was folded along the vertical and horizontal lines of symmetry (the mid - vertical and mid - horizontal lines), or in other words, the paper was folded in half vertically and then in half horizontally (or the other way around).
## Step2: Conclusion
The paper was folded along both the vertical and horizontal lines of symmetry (the mid - vertical and mid - horizontal lines), resulting in four layers, and punching a hole creates four holes as in figure (d).
# Answer:
The paper was folded along both the vertical and horizontal lines of symmetry (the mid - vertical and mid - horizontal lines), or equivalently, folded in half vertically and then in half horizontally (or horizontally first then vertically).

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QUESTION IMAGE

Question

Explanation:

Question 9

Step1: Find area of the wall

Step2: Find area of the door

Step3: Find area of the window

Step4: Subtract door and window areas from wall area

Step1: Identify the line of symmetry

Step2: Draw the reflected part

Step1: Recall rotational symmetry

Step2: Conclusion

Answer:

Question 10