Answered

The vertices of quadrilateral [tex]\(EFGH\)[/tex] are [tex]\(E(-2,-1), F(1,2), G(6,0)\)[/tex], and [tex]\(H(2,-2)\)[/tex].

Find the scale factor if an image of [tex]\(EFGH\)[/tex] has vertices [tex]\(E'\left(-3,-\frac{3}{2}\right), F'\left(\frac{3}{2}, 3\right), G'(9,0)\)[/tex], and [tex]\(H'(3,-3)\)[/tex].



Answer :

GinnyAnswer
To determine the scale factor for the transformation of quadrilateral [tex]\( EFGH \)[/tex] to [tex]\( E'F'G'H' \)[/tex], we need to compare the coordinates of the original vertices with those of the transformed vertices. The scale factor can be determined by calculating how each coordinate in the original shape is scaled.

Given:
- Coordinates of [tex]\( EFGH \)[/tex]:
- [tex]\( E(-2, -1) \)[/tex]
- [tex]\( F(1, 2) \)[/tex]
- [tex]\( G(6, 0) \)[/tex]
- [tex]\( H(2, -2) \)[/tex]
- Coordinates of [tex]\( E'F'G'H' \)[/tex]:
- [tex]\( E'(-3, -\frac{3}{2}) \)[/tex]
- [tex]\( F'(\frac{3}{2}, 3) \)[/tex]
- [tex]\( G'(9, 0) \)[/tex]
- [tex]\( H'(3, -3) \)[/tex]

### Step-by-Step Calculation:

1. Calculate the scale factor for point [tex]\( E \)[/tex] to [tex]\( E' \)[/tex]:
- For the [tex]\( x \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } x\text{-direction} = \frac{-3}{-2} = \frac{3}{2} \][/tex]
- For the [tex]\( y \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } y\text{-direction} = \frac{-\frac{3}{2}}{-1} = \frac{3}{2} \][/tex]

2. Calculate the scale factor for point [tex]\( F \)[/tex] to [tex]\( F' \)[/tex]:
- For the [tex]\( x \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } x\text{-direction} = \frac{\frac{3}{2}}{1} = \frac{3}{2} \][/tex]
- For the [tex]\( y \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } y\text{-direction} = \frac{3}{2} = \frac{3}{2} \][/tex]

3. Calculate the scale factor for point [tex]\( G \)[/tex] to [tex]\( G' \)[/tex]:
- For the [tex]\( x \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } x\text{-direction} = \frac{9}{6} = \frac{3}{2} \][/tex]
- For the [tex]\( y \)[/tex]-coordinates ([tex]\( y \)[/tex]-coordinate of [tex]\( G \)[/tex] and [tex]\( G' \)[/tex] is 0, so it stays 0):
[tex]\[ \text{Scale factor in the } y\text{-direction} = 0 ~ (\text{undefined as direction remains 0}) \][/tex]

4. Calculate the scale factor for point [tex]\( H \)[/tex] to [tex]\( H' \)[/tex]:
- For the [tex]\( x \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } x\text{-direction} = \frac{3}{2} = \frac{3}{2} \][/tex]
- For the [tex]\( y \)[/tex]-coordinates:
[tex]\[ \text{Scale factor in the } y\text{-direction} = \frac{-3}{-2} = \frac{3}{2} \][/tex]

### Conclusion

We can see from above that the new coordinates are consistently scaled by a factor of [tex]\( \frac{3}{2} \)[/tex] in both the [tex]\( x \)[/tex]-direction and [tex]\( y \)[/tex]-direction (except for the [tex]\( y\)[/tex]-coordinates of [tex]\( G \)[/tex] and [tex]\( G' \)[/tex]).

Thus, the scale factor for the transformation of quadrilateral [tex]\( EFGH \)[/tex] to [tex]\( E'F'G'H' \)[/tex] is:

[tex]\[ \boxed{\frac{3}{2}} \][/tex]