Two triangles, ABC and DEF, are similar. Side AB = 12 cm, BC = 9 cm, and AC = 15 cm. Side DE = 8 cm. Find the scale factor of reduction from ABC to DEF, and the lengths of sides EF (y) and DF (z).

The side AB (12 cm) in Triangle ABC corresponds to side DE (8 cm) in Triangle DEF.

$$\text{Scale factor of reduction} = \frac{\text{Length of DE}}{\text{Length of AB}} = \frac{8 \text{ cm}}{12 \text{ cm}}$$ $$\text{Scale factor of reduction} = \frac{8}{12} = \frac{2}{3}$$

The scale factor of reduction is $\boxed{\frac{2}{3}}$.

The side BC (9 cm) in Triangle ABC corresponds to side EF ($y$ cm) in Triangle DEF.

$$\frac{\text{EF}}{\text{BC}} = \text{Scale factor}$$ $$\frac{y}{9 \text{ cm}} = \frac{2}{3}$$ $$y = \frac{2}{3} \times 9 \text{ cm}$$ $$y = 2 \times 3 \text{ cm}$$ $$y = 6 \text{ cm}$$

The side AC (15 cm) in Triangle ABC corresponds to side DF ($z$ cm) in Triangle DEF.

$$\frac{\text{DF}}{\text{AC}} = \text{Scale factor}$$ $$\frac{z}{15 \text{ cm}} = \frac{2}{3}$$ $$z = \frac{2}{3} \times 15 \text{ cm}$$ $$z = 2 \times 5 \text{ cm}$$ $$z = 10 \text{ cm}$$

The value of $y$ is $\boxed{\text{6 cm}}$ and the value of $z$ is $\boxed{\text{10 cm}}$.

$$\text{Ratio of areas} = (\text{Scale factor})^2$$ $$\text{Ratio of areas} = \left(\frac{2}{3}\right)^2$$ $$\text{Ratio of areas} = \frac{2^2}{3^2} = \frac{4}{9}$$

The ratio of the area of Triangle DEF to the area of Triangle ABC is $\boxed{\frac{4}{9}}$.