Question

Polygon ABCD with vertices at A(1, −2), B(3, −2), C(3, −4), and D(1, −4) is dilated to create polygon A′B′C′D′ with vertices at A′(4, −8), B′(12, −8), C′(12, −16), and D′(4, −16). Determine the scale factor used to create the image.

one fourth
one half
2
4

Answer 1

To determine the scale factor used to dilate polygon ABCD to create polygon A′B′C′D′, we need to compare the corresponding coordinates of the original vertices to the dilated vertices.

Let's take the vertices of both polygons:

- A(1, -2) dilates to A′(4, -8)
- B(3, -2) dilates to B′(12, -8)
- C(3, -4) dilates to C′(12, -16)
- D(1, -4) dilates to D′(4, -16)

We can calculate the scale factor by comparing any corresponding coordinates.

Let's compare points A and A′:

\[
\text{Scale factor} = \frac{\text{Coordinate of A′}}{\text{Coordinate of A}}
\]

For the x-coordinates of A and A′:

\[
\text{Scale factor for x-coordinates} = \frac{4}{1} = 4
\]

For the y-coordinates of A and A′:

\[
\text{Scale factor for y-coordinates} = \frac{-8}{-2} = 4
\]

Both comparisons yield a scale factor of 4.

To verify this with other points:

For B and B′:

\[
\text{Scale factor for x-coordinates} = \frac{12}{3} = 4 \\
\text{Scale factor for y-coordinates} = \frac{-8}{-2} = 4
\]

For C and C′:

\[
\text{Scale factor for x-coordinates} = \frac{12}{3} = 4 \\
\text{Scale factor for y-coordinates} = \frac{-16}{-4} = 4
\]

For D and D′:

\[
\text{Scale factor for x-coordinates} = \frac{4}{1} = 4 \\
\text{Scale factor for y-coordinates} = \frac{-16}{-4} = 4
\]

Since all corresponding vertices give the same scale factor of 4, we conclude that the scale factor used to dilate the polygon is:

**4**.