Question
The vertices of trapezoid EFGH are located at E (1, 1), F (3, 1), G (4, -2), and H (0, -2). After EFGH has been dilated with a scale factor of 3 about the origin, which two statements are true about the vertices of image E'F'G'H' ?
The coordinates of E' are (1, 3).
The coordinates of G' are (7, 1).
The coordinates of H' are (0, -6).
Vertex F' is located in Quadrant I.
Vertex G' is located in Quadrant II.
The coordinates of E' are (1, 3).
The coordinates of G' are (7, 1).
The coordinates of H' are (0, -6).
Vertex F' is located in Quadrant I.
Vertex G' is located in Quadrant II.
Answers
GPT-4o mini
To perform a dilation around the origin with a scale factor of 3, we multiply each coordinate of the vertices of trapezoid EFGH by 3.
Let's calculate the new coordinates for each vertex:
1. **E** (1, 1) after dilation:
\[
E' = (1 \times 3, 1 \times 3) = (3, 3)
\]
2. **F** (3, 1) after dilation:
\[
F' = (3 \times 3, 1 \times 3) = (9, 3)
\]
3. **G** (4, -2) after dilation:
\[
G' = (4 \times 3, -2 \times 3) = (12, -6)
\]
4. **H** (0, -2) after dilation:
\[
H' = (0 \times 3, -2 \times 3) = (0, -6)
\]
Now, if we summarize the new coordinates of the vertices E', F', G', and H':
- \( E' = (3, 3) \)
- \( F' = (9, 3) \)
- \( G' = (12, -6) \)
- \( H' = (0, -6) \)
Now let's evaluate the statements:
1. **The coordinates of E' are (1, 3)**: **False**. The correct coordinates of E' are (3, 3).
2. **The coordinates of G' are (7, 1)**: **False**. The correct coordinates of G' are (12, -6).
3. **The coordinates of H' are (0, -6)**: **True**. This is correct.
4. **Vertex F' is located in Quadrant I**: **True**. The coordinates (9, 3) place F' in Quadrant I.
5. **Vertex G' is located in Quadrant II**: **False**. The coordinates (12, -6) place G' in Quadrant IV, not Quadrant II.
Based on this analysis, the two true statements about the vertices of image E'F'G'H' are:
- The coordinates of H' are (0, -6).
- Vertex F' is located in Quadrant I.
Let's calculate the new coordinates for each vertex:
1. **E** (1, 1) after dilation:
\[
E' = (1 \times 3, 1 \times 3) = (3, 3)
\]
2. **F** (3, 1) after dilation:
\[
F' = (3 \times 3, 1 \times 3) = (9, 3)
\]
3. **G** (4, -2) after dilation:
\[
G' = (4 \times 3, -2 \times 3) = (12, -6)
\]
4. **H** (0, -2) after dilation:
\[
H' = (0 \times 3, -2 \times 3) = (0, -6)
\]
Now, if we summarize the new coordinates of the vertices E', F', G', and H':
- \( E' = (3, 3) \)
- \( F' = (9, 3) \)
- \( G' = (12, -6) \)
- \( H' = (0, -6) \)
Now let's evaluate the statements:
1. **The coordinates of E' are (1, 3)**: **False**. The correct coordinates of E' are (3, 3).
2. **The coordinates of G' are (7, 1)**: **False**. The correct coordinates of G' are (12, -6).
3. **The coordinates of H' are (0, -6)**: **True**. This is correct.
4. **Vertex F' is located in Quadrant I**: **True**. The coordinates (9, 3) place F' in Quadrant I.
5. **Vertex G' is located in Quadrant II**: **False**. The coordinates (12, -6) place G' in Quadrant IV, not Quadrant II.
Based on this analysis, the two true statements about the vertices of image E'F'G'H' are:
- The coordinates of H' are (0, -6).
- Vertex F' is located in Quadrant I.
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