https://file.aiquickdraw.com/mathbot/file/1726311473542ifa9rsxs.jpg

Let's go step-by-step through the problem based on the image you provided:

### 1. **Plot points A, B, and C on the grid and join them**:
   - **A = (1, 1)**
   - **B = (3, 3)**
   - **C = (4, 1)**

After plotting these points on the given grid, you can connect them to form triangle **ABC**.

### 2. **Transformations on triangle ABC**:

#### a) **Translate by \(\begin{pmatrix} 2 \\ 1 \end{pmatrix}\)**:
   - Translation involves shifting the triangle. 
     - For point A: \((1, 1) \rightarrow (1 + 2, 1 + 1) = (3, 2)\)
     - For point B: \((3, 3) \rightarrow (3 + 2, 3 + 1) = (5, 4)\)
     - For point C: \((4, 1) \rightarrow (4 + 2, 1 + 1) = (6, 2)\)
   - New coordinates of triangle **A'B'C'** after translation are: **A'(3, 2), B'(5, 4), C'(6, 2)**.

#### b) **Reflect in the y-axis**:
   - Reflection in the y-axis changes the sign of the x-coordinate.
     - For point A: \((1, 1) \rightarrow (-1, 1)\)
     - For point B: \((3, 3) \rightarrow (-3, 3)\)
     - For point C: \((4, 1) \rightarrow (-4, 1)\)
   - New coordinates of triangle **A"B"C"** after reflection are: **A"(-1, 1), B"(-3, 3), C"(-4, 1)**.

#### c) **Rotate clockwise by 180° about (0, 0)**:
   - Rotation by 180° changes both the x- and y-coordinates to their opposites.
     - For point A: \((1, 1) \rightarrow (-1, -1)\)
     - For point B: \((3, 3) \rightarrow (-3, -3)\)
     - For point C: \((4, 1) \rightarrow (-4, -1)\)
   - New coordinates of triangle **A'''B'''C'''** after 180° rotation are: **A'''(-1, -1), B'''(-3, -3), C'''(-4, -1)**.

#### d) **Reflect in the line \(y = -1\)**:
   - To reflect across the line \(y = -1\), calculate the distance from each point to \(y = -1\), then move each point the same distance in the opposite direction.
     - For point A: The distance from \((1, 1)\) to \(y = -1\) is 2 units. So, after reflection, \(A \rightarrow (1, -3)\).
     - For point B: The distance from \((3, 3)\) to \(y = -1\) is 4 units. So, after reflection, \(B \rightarrow (3, -5)\).
     - For point C: The distance from \((4, 1)\) to \(y = -1\) is 2 units. So, after reflection, \(C \rightarrow (4, -3)\).
   - New coordinates of triangle **A''''B''''C''''** after reflection are: **A''''(1, -3), B''''(3, -5), C''''(4, -3)**.

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Let me know if you'd like any further details on the steps or have additional questions.

Here are five relative questions:
1. How do you plot points on a coordinate plane?
2. What is the general rule for translating points in a coordinate plane?
3. What happens to the coordinates of a point when reflecting in the y-axis?
4. How do you calculate a 180° rotation of a point around the origin?
5. How can you determine the reflection of a point across a horizontal line like \(y = -1\)?

### Tip:
When reflecting over a line, first calculate the perpendicular distance from the point to the line. Then move the same distance in the opposite direction!

Plot points A to C and perform various transformations on triangle ABC, including translation, reflection, and rotation.

Learn how to perform a series of transformations on triangle ABC, including translation by (2,1), reflection in the y-axis, rotation by 180°, and reflection in the line y = -1. Detailed explanations of each step provided for students in Grades 9-10.

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