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. \[ \triangle \text { TOOLS } \]

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.