Math / Geometry

QuestionGraph $\triangle F G H$ with vertices $F(-2,2), G(-2,-4)$ and $H(-4,-4)$ and its image after the similarity transformation. Dilation: $(x, y) \rightarrow\left(\frac{1}{2} x, \frac{1}{2} y\right)$ Reflection: in the $y$ -axis Polygon Undo Redo Reset

Studdy Solution

STEP 1

1. The vertices of $\triangle FGH$ are given as $F(-2, 2)$ , $G(-2, -4)$ , and $H(-4, -4)$ .
2. The similarity transformation involves a dilation followed by a reflection.
3. The dilation transformation is $(x, y) \rightarrow \left(\frac{1}{2} x, \frac{1}{2} y\right)$ .
4. The reflection is across the $y$ -axis.

STEP 2

1. Apply the dilation transformation to each vertex.
2. Apply the reflection transformation to each vertex.
3. Graph the original triangle and its image.

STEP 3

Apply the dilation transformation $(x, y) \rightarrow \left(\frac{1}{2} x, \frac{1}{2} y\right)$ to each vertex:
- For $F(-2, 2)$ : $ F' = \left(\frac{1}{2} \times -2, \frac{1}{2} \times 2\right) = (-1, 1) \]
- For $G(-2, -4)$ : $ G' = \left(\frac{1}{2} \times -2, \frac{1}{2} \times -4\right) = (-1, -2) \]
- For $H(-4, -4)$ : $ H' = \left(\frac{1}{2} \times -4, \frac{1}{2} \times -4\right) = (-2, -2) \]

STEP 4

Apply the reflection transformation across the $y$ -axis to each vertex:
- For $F'(-1, 1)$ : $ F'' = (1, 1) \]
- For $G'(-1, -2)$ : $ G'' = (1, -2) \]
- For $H'(-2, -2)$ : $ H'' = (2, -2) \]

STEP 5

Graph the original triangle $\triangle FGH$ with vertices $F(-2, 2)$ , $G(-2, -4)$ , and $H(-4, -4)$ .
Graph the image of the triangle $\triangle F''G''H''$ with vertices $F''(1, 1)$ , $G''(1, -2)$ , and $H''(2, -2)$ .
The transformation results in the image $\triangle F''G''H''$ with vertices $F''(1, 1)$ , $G''(1, -2)$ , and $H''(2, -2)$ .

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