To solve this problem, let's go through each of the transformations step-by-step, starting with the initial coordinates of points $A$, $B$, and $C$.

Step 1: Identifying the Initial Coordinates

From the image:

• $A$ is located at approximately $(0, -2)$
• $B$ is located at approximately $(2, 1)$
• $C$ is located at approximately $(3, -1)$

Let's proceed with each transformation in sequence.

Step 2: Rotation 180 Degrees About the Origin

To rotate a point $(x, y)$ by 180 degrees around the origin, the coordinates become $(-x, -y)$.

Applying this to each point:

• $A(0, -2)$ becomes $A' (0, 2)$
• $B(2, 1)$ becomes $B' (-2, -1)$
• $C(3, -1)$ becomes $C' (-3, 1)$

New coordinates after rotation:

• $A' = (0, 2)$
• $B' = (-2, -1)$
• $C' = (-3, 1)$

Step 3: Reflection Over the X-axis

To reflect a point $(x, y)$ over the x-axis, the coordinates become $(x, -y)$.

Applying this to each rotated point:

• $A'(0, 2)$ becomes $A'' (0, -2)$
• $B'(-2, -1)$ becomes $B'' (-2, 1)$
• $C'(-3, 1)$ becomes $C'' (-3, -1)$

New coordinates after reflection:

• $A'' = (0, -2)$
• $B'' = (-2, 1)$
• $C'' = (-3, -1)$

Step 4: Translation Up 1 Unit and Right 3 Units

To translate a point $(x, y)$ by moving it right 3 units and up 1 unit, the new coordinates become $(x + 3, y + 1)$.

Applying this to each reflected point:

• $A''(0, -2)$ becomes $A''' (0 + 3, -2 + 1) = (3, -1)$
• $B''(-2, 1)$ becomes $B''' (-2 + 3, 1 + 1) = (1, 2)$
• $C''(-3, -1)$ becomes $C''' (-3 + 3, -1 + 1) = (0, 0)$

Final Coordinates of $A'''B'''C'''$:

• $A''' = (3, -1)$
• $B''' = (1, 2)$
• $C''' = (0, 0)$

Would you like further clarification on any of these steps, or do you have other questions?

Related Questions

1. What are the general rules for rotating points by 90°, 180°, and 270° about the origin?
2. How do transformations combine when performed in a different order?
3. What are the coordinates of triangle $A'''B'''C'''$ if only a reflection over the y-axis was applied?
4. How do you find the image of a point after a sequence of multiple transformations?
5. How would this change if the translation was different, such as moving left instead of right?

Tip

When performing multiple transformations, always keep track of each transformation's impact on coordinates step-by-step to avoid mistakes.