Question

graph the image of $\triangle ghi$ after the following sequence of transformations:
rotation $180^\circ$ counterclockwise around the origin
translation 1 unit left and 21 units down

Explanation:

First, we need to identify the coordinates of the vertices of $\triangle GHI$. From the graph, we can see that:

• $G$ is at $(8, -14)$
• $H$ is at $(6, -8)$
• $I$ is at $(2, -14)$

Step 1: Rotation 180° counterclockwise around the origin

The rule for a 180° counterclockwise rotation around the origin is $(x, y) \to (-x, -y)$.

• For $G(8, -14)$: $(-8, 14)$
• For $H(6, -8)$: $(-6, 8)$
• For $I(2, -14)$: $(-2, 14)$

Step 2: Translation 1 unit left and 21 units down

The rule for a translation 1 unit left (subtract 1 from the x - coordinate) and 21 units down (subtract 21 from the y - coordinate) is $(x, y) \to (x - 1, y - 21)$.

• For the rotated $G(-8, 14)$: $(-8 - 1, 14 - 21)=(-9, -7)$
• For the rotated $H(-6, 8)$: $(-6 - 1, 8 - 21)=(-7, -13)$
• For the rotated $I(-2, 14)$: $(-2 - 1, 14 - 21)=(-3, -7)$

Now we can plot the points $(-9, -7)$, $(-7, -13)$ and $(-3, -7)$ to get the image of $\triangle GHI$ after the transformations.

Answer:

The vertices of the image of $\triangle GHI$ are $G'(-9, -7)$, $H'(-7, -13)$ and $I'(-3, -7)$. To graph the image, plot these three points and connect them to form the triangle.