Question: 5. Here is triangle $XYZ$: Draw these three rotations of triangle $XYZ$ together. a. Rotate triangle $XYZ$ 90 degrees clockwise around $Z$. b. Rotate triangle $XYZ$ 180 degrees around $Z$. c. Rotate triangle $XYZ$ 270 degrees clockwise around $Z$. (From Unit 1, Lesson 8.) Graph/shape description: A triangle with vertices $Z$ at upper-left, $Y$ at lower-left, and $X$ at right; segment $ZY$ is vertical, and segments $ZX$ and $YX$ extend to the right to form a narrow triangle. position_hint=top-right

1. **Problem Statement:** We have triangle $XYZ$ with vertices $Z$ at upper-left, $Y$ at lower-left, and $X$ at right. We need to draw three rotations of this triangle around point $Z$: - a. Rotate $90^\circ$ clockwise around $Z$. - b. Rotate $180^\circ$ around $Z$. - c. Rotate $270^\circ$ clockwise around $Z$. 2. **Rotation Rules:** - Rotations are performed around a fixed point, here $Z$. - A $90^\circ$ clockwise rotation moves a point $(x,y)$ relative to $Z$ to $(y,-x)$. - A $180^\circ$ rotation moves $(x,y)$ to $(-x,-y)$. - A $270^\circ$ clockwise rotation is equivalent to a $90^\circ$ counterclockwise rotation, moving $(x,y)$ to $(-y,x)$. 3. **Assign Coordinates:** Assume $Z$ is at the origin $(0,0)$ for simplicity. Let $Y$ be at $(0,-a)$ since $ZY$ is vertical downward. Let $X$ be at $(b,c)$ with $b>0$ and $c$ near zero to the right. 4. **Apply Rotations:** - a. $90^\circ$ clockwise rotation around $Z$: - $Y(0,-a) \to ( -a, 0 )$ - $X(b,c) \to ( c, -b )$ - b. $180^\circ$ rotation around $Z$: - $Y(0,-a) \to (0,a)$ - $X(b,c) \to (-b,-c)$ - c. $270^\circ$ clockwise rotation around $Z$: - $Y(0,-a) \to (a,0)$ - $X(b,c) \to (-c,b)$ 5. **Summary of Rotated Vertices:** | Rotation | $Y'$ | $X'$ | |---|---|---| | $90^\circ$ CW | $(-a,0)$ | $(c,-b)$ | | $180^\circ$ | $(0,a)$ | $(-b,-c)$ | | $270^\circ$ CW | $(a,0)$ | $(-c,b)$ | 6. **Interpretation:** Each rotation moves the triangle around $Z$ by the specified angle, changing the positions of $X$ and $Y$ accordingly. This completes the rotations as requested.