The vertices of a figure are $$ G(-2,-2), H(-2,6), J(2,6) ; k=0.25 $$. Draw the tigure and its image amter a diabuon with th given scale factor. $$ \triangle \text { TOOLS } $$

Question

UpStudy AI · Accepted Answer

To perform a **dilation** (also known as a scaling transformation) on the given triangle with a **scale factor** $$ k = 0.25 $$, we'll follow these steps:

1. **Plot the Original Triangle** with vertices $$ G(-2, -2) $$, $$ H(-2, 6) $$, and $$ J(2, 6) $$.
2. **Apply the Dilation** to each vertex to find the coordinates of the image triangle.
3. **Plot the Image Triangle** to visualize the transformation.

Since we cannot provide a graphical drawing here, I'll illustrate the process using coordinates and describe the positions of both triangles on a coordinate plane.

---

### 1. Original Triangle Coordinates

- **Vertex G**: $$(-2, -2)$$
- **Vertex H**: $$(-2, 6)$$
- **Vertex J**: $$(2, 6)$$

**Visualization on Coordinate Plane:**

```
Y
|
6 |        H (-2,6) ________ J (2,6)
  |          |               |
  |          |               |
  |          |               |
  |          |               |
0 |          |               |
  |          |               |
  |          |               |
  |          |               |
-2|__________G (-2,-2)        |
  |______________________________ X
        -2        0          2
```

- **Side GH** is vertical from $$ (-2, -2) $$ to $$ (-2, 6) $$.
- **Side HJ** is horizontal from $$ (-2, 6) $$ to $$ (2, 6) $$.
- **Side JG** connects $$ (2, 6) $$ back to $$ (-2, -2) $$, forming a right triangle.

---

### 2. Applying Dilation with Scale Factor $$ k = 0.25 $$

**Dilation Formula:**
$$
(x', y') = (k \cdot x, k \cdot y)
$$

**Calculations for Each Vertex:**

- **Vertex G':**
  $$
  G' = (0.25 \times -2, 0.25 \times -2) = (-0.5, -0.5)
  $$
  
- **Vertex H':**
  $$
  H' = (0.25 \times -2, 0.25 \times 6) = (-0.5, 1.5)
  $$
  
- **Vertex J':**
  $$
  J' = (0.25 \times 2, 0.25 \times 6) = (0.5, 1.5)
  $$

**Image Triangle Coordinates:**

- **Vertex G'**: $$(-0.5, -0.5)$$
- **Vertex H'**: $$(-0.5, 1.5)$$
- **Vertex J'**: $$(0.5, 1.5)$$

---

### 3. Image Triangle Visualization

```
Y
|
1.5 |        H' (-0.5,1.5) ______ J' (0.5,1.5)
    |           |               |
    |           |               |
    |           |               |
0.5 |           |               |
    |           |               |
    |           |               |
    |           |               |
-0.5|___________G' (-0.5,-0.5)    |
    |______________________________ X
           -0.5         0         0.5
```

- **Side G'H'** is vertical from $$ (-0.5, -0.5) $$ to $$ (-0.5, 1.5) $$.
- **Side H'J'** is horizontal from $$ (-0.5, 1.5) $$ to $$ (0.5, 1.5) $$.
- **Side J'G'** connects $$ (0.5, 1.5) $$ back to $$ (-0.5, -0.5) $$, forming a smaller right triangle.

---

### Summary of Transformation

- **Original Triangle**:
  - **Vertices**: $$ G(-2, -2) $$, $$ H(-2, 6) $$, $$ J(2, 6) $$
  - **Shape**: Right triangle with legs of lengths 8 (vertical) and 4 (horizontal).

- **Dilated Image Triangle**:
  - **Vertices**: $$ G'(-0.5, -0.5) $$, $$ H'(-0.5, 1.5) $$, $$ J'(0.5, 1.5) $$
  - **Shape**: Similar right triangle scaled down by a factor of 0.25, resulting in legs of lengths 2 (vertical) and 1 (horizontal).

**Note:** The dilation is performed with respect to the **origin (0,0)**. If a different center of dilation is desired, the process would need to adjust accordingly.

---

### Tools for Visualization

To visualize these triangles graphically, you can use graphing software or online tools such as:

- **Desmos**: [https://www.desmos.com/calculator](https://www.desmos.com/calculator)
- **GeoGebra**: [https://www.geogebra.org/graphing](https://www.geogebra.org/graphing)
- **Microsoft Excel** or **Google Sheets**: Using scatter plots.

**Example Steps in Desmos:**

1. **Plot Original Triangle:**
   - Enter points: $$ G(-2, -2) $$, $$ H(-2, 6) $$, $$ J(2, 6) $$.
   - Draw segments between them.

2. **Plot Image Triangle:**
   - Enter points: $$ G'(-0.5, -0.5) $$, $$ H'(-0.5, 1.5) $$, $$ J'(0.5, 1.5) $$.
   - Draw segments between them.

3. **Compare Both Triangles:**
   - Observe the scaling effect of the dilation.

---

If you have access to these tools, you can create a visual representation to better understand the dilation transformation.
To perform a dilation with a scale factor of 0.25 on the triangle with vertices $$ G(-2,-2) $$, $$ H(-2,6) $$, and $$ J(2,6) $$, follow these steps: 1. **Original Triangle:** - $$ G(-2, -2) $$ - $$ H(-2, 6) $$ - $$ J(2, 6) $$ 2. **Dilated Image Triangle:** - $$ G'(-0.5, -0.5) $$ - $$ H'(-0.5, 1.5) $$ - $$ J'(0.5, 1.5) $$ **Visualization:** - Plot both triangles on a coordinate plane to see the scaling effect. This dilation reduces the size of the original triangle by a factor of 0.25.

The vertices of a figure are . Draw the tigure and its image amter a diabuon with th given scale factor.

Solución de inteligencia artificial de Upstudy

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Solución

1. Original Triangle Coordinates

2. Applying Dilation with Scale Factor

3. Image Triangle Visualization

Summary of Transformation

Tools for Visualization

Bonus Knowledge

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