examine the figure graphed on the axes at right. 1-108 hw etool homework help a. what happens when you rotate this figure about the origin 90°? 45°? 180°? b. what other angle could the figure above be rotated so the shape does not appear to change? c. what shape will stay the same no matter how many degrees it is rotated?

Question

Accepted Answer

### Part a
#### Explanation:
To determine the effect of rotating the given figure (which is symmetric about both the x - axis, y - axis and has rotational symmetry) about the origin:

- **Step 1: Rotation by $90^\circ$ about the origin**
A rotation of $90^\circ$ about the origin follows the rule $(x,y)\to(-y,x)$ (for counter - clockwise rotation) or $(x,y)\to(y, - x)$ (for clockwise rotation). The given figure has a symmetry such that it is symmetric with respect to the x - axis and y - axis but not with respect to a $90^\circ$ rotation. So, when we rotate the figure by $90^\circ$ (either clockwise or counter - clockwise) about the origin, the figure will not coincide with its original position. The shape will look different from the original.
- **Step 2: Rotation by $45^\circ$ about the origin**
A $45^\circ$ rotation about the origin has the transformation rule $(x,y)\to(x\cos45^{\circ}-y\sin45^{\circ},x\sin45^{\circ}+y\cos45^{\circ})$ (for counter - clockwise) or $(x,y)\to(x\cos(- 45^{\circ})-y\sin(- 45^{\circ}),x\sin(- 45^{\circ})+y\cos(- 45^{\circ}))$ (for clockwise). Since the figure does not have $45^\circ$ rotational symmetry (it has a more "square - like" symmetry with respect to $180^\circ$ and $180^\circ$ multiples), rotating it by $45^\circ$ will result in a figure that does not match the original. The shape will be in a different orientation and will not appear the same.
- **Step 3: Rotation by $180^\circ$ about the origin**
The rule for a $180^\circ$ rotation about the origin is $(x,y)\to(-x,-y)$. The given figure is symmetric about the origin (since it is symmetric with respect to both x - axis and y - axis, and the composition of a reflection over x - axis and y - axis is a $180^\circ$ rotation). So, when we rotate the figure by $180^\circ$ about the origin, the figure will coincide with its original position. The shape will appear to be the same as the original.

### Part b
#### Explanation:
- **Step 1: Analyze rotational symmetry**
The figure has rotational symmetry. We know that for a figure with rotational symmetry, the angle of rotation $\theta$ satisfies $\theta=\frac{360^\circ}{n}$, where $n$ is a positive integer (the order of rotational symmetry). We saw that a $180^\circ$ rotation works. Let's check the order of rotational symmetry. If we rotate the figure by $180^\circ$ it maps onto itself. Also, if we rotate it by $180^\circ\times k$ where $k = 1,2,\cdots$ it will map onto itself. But we can also consider that the figure is symmetric with respect to rotations of $180^\circ$ and its multiples. Another angle could be $180^\circ\times2 = 360^\circ$ (a full rotation, which maps any figure onto itself), but also, since the figure is symmetric about the origin (central symmetry), any rotation by $180^\circ$ plus a multiple of $180^\circ$ will work. But a non - trivial angle (other than $180^\circ$ and $360^\circ$)? Wait, no, the figure as drawn (symmetric about x and y axes) has rotational symmetry of order 2 (since $360^\circ\div2=180^\circ$). Wait, maybe I made a mistake. Wait, the figure looks like a shape with 4 - fold? No, looking at the figure, it is symmetric with respect to x - axis, y - axis, and origin. So when we rotate it by $180^\circ$ it matches. Also, if we rotate it by $180^\circ$, and also, what about $ - 180^\circ$ (which is the same as $180^\circ$ in terms of rotation). Wait, maybe the figure has rotational symmetry of order 2, so the angle of rotation is $180^\circ$, and also $180^\circ + 180^\circ=360^\circ$. But also, if we consider that the figure is symmetric, another angle could be $180^\circ$ (we already considered that), wait, no. Wait, the question is "what other angle could the figure above be rotated so the shape does not appear to change?". Since it has $180^\circ$ rotational symmetry, any rotation by $180^\circ\times k$, where $k\in\mathbb{Z}$ (integers). So $180^\circ$ (we saw that), $360^\circ$ (a full rotation), $- 180^\circ$ (same as $180^\circ$). But maybe the figure is symmetric with respect to $90^\circ$? No, the figure as drawn (with the "points" on the axes) - if we rotate by $90^\circ$, the points on the y - axis will go to the x - axis, but the shape of the "arms" - maybe not. Wait, maybe the figure has rotational symmetry of order 2, so the angle is $180^\circ$, and also, if we rotate by $180^\circ$, it works. Wait, maybe the answer is $180^\circ$ (but we already considered that in part a). Wait, no, part a asks about $90^\circ$, $45^\circ$, $180^\circ$, part b asks for another angle. Wait, maybe the figure is symmetric with respect to $180^\circ$ rotation, so another angle could be $180^\circ$ (but that's in part a). Wait, no, maybe I mis - analyzed the figure. Let's think again. The figure is symmetric about the x - axis, y - axis, and origin. So when you rotate it by $180^\circ$, it maps to itself. Also, if you rotate it by $0^\circ$ (no rotation) it maps to itself, $360^\circ$ (same as $0^\circ$). But the question says "other angle" (other than what? Maybe other than the ones in part a). Wait, in part a, we considered $90^\circ$, $45^\circ$, $180^\circ$. So another angle could be $180^\circ$ (but that's in part a). Wait, no, maybe the figure has rotational symmetry of order 2, so the angle is $180^\circ$, and also, if we rotate by $180^\circ$, it works. Alternatively, maybe the figure is a square - like shape with rotational symmetry of order 4? No, the figure has "points" on the x and y axes. Wait, maybe the answer is $180^\circ$ (but that's in part a). Wait, no, the question is "what other angle could the figure above be rotated so the shape does not appear to change?". Since it has $180^\circ$ rotational symmetry, any multiple of $180^\circ$, so $180^\circ$, $360^\circ$, $- 180^\circ$. But $360^\circ$ is a full rotation, so the shape will not appear to change. So the answer could be $360^\circ$ (or $- 180^\circ$ which is equivalent to $180^\circ$ in terms of the final position).

### Part c
#### Explanation:
- **Step 1: Recall shapes with full rotational symmetry**
A circle is a shape that has infinite rotational symmetry. This means that no matter what angle we rotate a circle about its center (which is like the origin for a circle centered at the origin), the circle will coincide with its original position. For any other shape (like a square, triangle, the given figure in the problem), there are only specific angles of rotation (rotational symmetry angles) for which the shape coincides with itself. A circle, when rotated by any number of degrees about its center, will look exactly the same because all points on the circle are equidistant from the center, and there is no "distinguishing" feature or orientation.

### Final Answers

#### Part a
- Rotation by $90^\circ$: The figure does not coincide with its original position (shape appears different).
- Rotation by $45^\circ$: The figure does not coincide with its original position (shape appears different).
- Rotation by $180^\circ$: The figure coincides with its original position (shape appears the same).

#### Part b
The figure can be rotated by $360^\circ$ (or any multiple of $180^\circ$ like $- 180^\circ$) so that the shape does not appear to change. (A more precise answer: $180^\circ$ is already considered, but $360^\circ$ is a full rotation and also works. Alternatively, if the figure has rotational symmetry of order 2, the angle is $180^\circ$, and another angle is $180^\circ\times2 = 360^\circ$)

#### Part c
A circle will stay the same no matter how many degrees it is rotated.

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QUESTION IMAGE

Question

Explanation:

Part a

Part b

Final Answers

Part a

Part b

Part c

Answer:

Final Answers

Part a

Part b

Part c