What are the properties of similar triangles related to angles, scale factors, and perimeters?

Scale factor of $\triangle A$ to $\triangle B$ is $\frac{4}{5}$.

The ratio of the perimeters of $\triangle A$ to $\triangle B$ is also $\frac{4}{5}$.

Answer: The ratio of the perimeters of $\triangle A$ to $\triangle B$ is $\boxed*{\frac{4}{5}*}$.

7. $\triangle ABC$ is similar to $\triangle DEF$. Find the perimeter of $\triangle DEF$.

Perimeter of $\triangle ABC = AB + BC + AC = 8 + 10 + 12 = 30$ cm.

We compare corresponding sides. Side $AC$ corresponds to side $DF$, and side $AB$ corresponds to side $DE$.
Using sides $AB$ and $DE$: Scale factor $k = \frac{DE}{AB} = \frac{24}{8} = 3$.
Using sides $AC$ and $DF$: Scale factor $k = \frac{DF}{AC} = \frac{36}{12} = 3$. (Assuming $DF=36$ based on the scale factor, as $DF$ is not explicitly labeled but $DE=24$ is).

Since the triangles are similar, the ratio of their perimeters is equal to the scale factor.
Perimeter of $\triangle DEF = \text{Scale factor} \times \text{Perimeter of } \triangle ABC$
Perimeter of $\triangle DEF = 3 \times 30 \text{ cm}$
Perimeter of $\triangle DEF = 90 \text{ cm}$

Answer: The perimeter of $\triangle DEF$ is $\boxed{90 \text{ cm}}$.

8. A cube has all sides of length 2 cm. What will be the area of the hexagon be, if it is enlarged with scale factor 3?
This question appears to have a slight mix-up in wording, referring to a cube but then asking about a hexagon. Based on the image, we will find the area of the enlarged hexagon.

Original area of the hexagon = $60 \text{ cm}^2$.
Scale factor $k = 3$.

If a figure is enlarged by a scale factor $k$, its area is enlarged by a factor of $k^2$.
New Area = Original Area $\times k^2$
New Area = $60 \text{ cm}^2 \times (3)^2$
New Area = $60 \text{ cm}^2 \times 9$
New Area = $540 \text{ cm}^2$

Answer: The area of the enlarged hexagon will be $\boxed{540 \text{ cm}^2}$.

9. What scale factor is needed to enlarge the smaller cuboid to become larger?
To find the scale factor, we compare the lengths of corresponding sides of the two cuboids.

Smaller cuboid: Length = 4 cm, Width = 3 cm, Height = 2 cm.
Larger cuboid: Length = 8 cm, Width = 6 cm, Height = 4 cm.

Ratio of lengths: $\frac{8 \text{ cm}}{4 \text{ cm}} = 2$
Ratio of widths: $\frac{6 \text{ cm}}{3 \text{ cm}} = 2$
Ratio of heights: $\frac{4 \text{ cm}}{2 \text{ cm}} = 2$

Since all ratios are equal, the scale factor is 2.

Answer: The scale factor is $\boxed{2}$.