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 determine the coordinates of the vertices of trapezoid EFGH after a dilation with a scale factor of 3 about the origin, we can apply the dilation transformation to each vertex. The formula for dilating a point \((x, y)\) by a scale factor \(k\) about the origin is given by:

\[
(x', y') = (kx, ky)
\]

Using this formula, let's calculate the new coordinates for each vertex:

1. **For vertex E (1, 1)**:
\[
E' = (3 \times 1, 3 \times 1) = (3, 3)
\]

2. **For vertex F (3, 1)**:
\[
F' = (3 \times 3, 3 \times 1) = (9, 3)
\]

3. **For vertex G (4, -2)**:
\[
G' = (3 \times 4, 3 \times -2) = (12, -6)
\]

4. **For vertex H (0, -2)**:
\[
H' = (3 \times 0, 3 \times -2) = (0, -6)
\]

Now, summarizing the coordinates of the vertices of the dilated trapezoid E'F'G'H':

- \(E'\) is (3, 3)
- \(F'\) is (9, 3)
- \(G'\) is (12, -6)
- \(H'\) is (0, -6)

Now, let's evaluate the given statements:

1. **The coordinates of E' are (1, 3).**
- **False**; the coordinates of \(E'\) are (3, 3).

2. **The coordinates of G' are (7, 1).**
- **False**; the coordinates of \(G'\) are (12, -6).

3. **The coordinates of H' are (0, -6).**
- **True**; the coordinates of \(H'\) are correct.

4. **Vertex F' is located in Quadrant I.**
- **True**; \(F' (9, 3)\) is located in Quadrant I.

5. **Vertex G' is located in Quadrant II.**
- **False**; \(G' (12, -6)\) is located in Quadrant IV.

Based on this analysis, the two true statements are:
- The coordinates of H' are (0, -6).
- Vertex F' is located in Quadrant I.