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.

Answer 1

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.