Question

The triangle D'E'F' is a dilation of the triangle DEF. What is the scale factor of the dilation?
D'(-1,-3)
E'(1,3)
F'(-2,3)
D(-3,-9)
E(3,9)
F(-6,9)
Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Answer 1

To find the scale factor of the dilation from triangle DEF to triangle D'E'F', we can use the coordinates of corresponding points. The scale factor \( k \) can be found using the formula:

\[
k = \frac{\text{distance from the origin to a point in D'E'F'}}{\text{distance from the origin to the corresponding point in DEF}}
\]

### Step 1: Calculate the distances from the origin (0,0) to each point.

1. **For point D(-3, -9)**:
\[
d_{D} = \sqrt{(-3)^2 + (-9)^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10}
\]

2. **For point D'(-1, -3)**:
\[
d_{D'} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]

3. **For point E(3, 9)**:
\[
d_{E} = \sqrt{(3)^2 + (9)^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10}
\]

4. **For point E'(1, 3)**:
\[
d_{E'} = \sqrt{(1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]

5. **For point F(-6, 9)**:
\[
d_{F} = \sqrt{(-6)^2 + (9)^2} = \sqrt{36 + 81} = \sqrt{117} = 3\sqrt{13}
\]

6. **For point F'(-2, 3)**:
\[
d_{F'} = \sqrt{(-2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13}
\]

### Step 2: Calculate the scale factors for each pairs of points.

Using points D and D':
\[
k_D = \frac{d_{D'}}{d_{D}} = \frac{\sqrt{10}}{3\sqrt{10}} = \frac{1}{3}
\]

Using points E and E':
\[
k_E = \frac{d_{E'}}{d_{E}} = \frac{\sqrt{10}}{3\sqrt{10}} = \frac{1}{3}
\]

Using points F and F':
\[
k_F = \frac{d_{F'}}{d_{F}} = \frac{\sqrt{13}}{3\sqrt{13}} = \frac{1}{3}
\]

### Conclusion

The scale factor of the dilation from triangle DEF to triangle D'E'F' is:

\[
\boxed{\frac{1}{3}}
\]