Question

he 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' ?

Answer 1

To determine the coordinates of the dilated trapezoid EFGH after a dilation with a scale factor of 3 about the origin, we can use the formula for dilation of a point \( (x, y) \) about the origin:

\[
(x, y) \rightarrow (kx, ky)
\]

where \( k \) is the scale factor.

Given the vertices of trapezoid EFGH:

- E (1, 1)
- F (3, 1)
- G (4, -2)
- H (0, -2)

Applying the dilation with a scale factor of 3:

1. **Vertex E**:
\[
E' = (3 \cdot 1, 3 \cdot 1) = (3, 3)
\]

2. **Vertex F**:
\[
F' = (3 \cdot 3, 3 \cdot 1) = (9, 3)
\]

3. **Vertex G**:
\[
G' = (3 \cdot 4, 3 \cdot -2) = (12, -6)
\]

4. **Vertex H**:
\[
H' = (3 \cdot 0, 3 \cdot -2) = (0, -6)
\]

Now we have the new vertices of the dilated trapezoid E'F'G'H':

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

### Statements about the vertices of image E'F'G'H':

1. **The lengths of the bases of the trapezoid E'F'G'H' are proportional to the original trapezoid EFGH.**
- Since dilation maintains the shape but scales the distance, if EFGH had bases EF and GH, E'F' will be a scaled version of EF, and likewise for GH.

2. **The trapezoid E'F'G'H' is similar to trapezoid EFGH.**
- All corresponding angles are preserved, and the ratios of corresponding sides remain constant, which is a property of similar figures.

Therefore, the two true statements about the vertices of image E'F'G'H' are that their corresponding sides are proportional and the trapezoids are similar.