We are asked to find the lateral area ($L$) and surface area ($S$) of a triangular prism. The base of the prism is a triangle with sides 9 cm, 9 cm, and 9 cm. The height of the prism is 12 cm. The height of the triangular base is 7.8 cm. We need to round to the nearest tenth, if necessary.

1. Problem Description

We are asked to find the lateral area ($L$) and surface area ($S$) of a triangular prism. The base of the prism is a triangle with sides 9 cm, 9 cm, and 9 cm. The height of the prism is 12 cm. The height of the triangular base is 7.8 cm. We need to round to the nearest tenth, if necessary.

2. Solution Steps

The lateral area of a prism is given by the formula:

$L = Ph$

where $P$ is the perimeter of the base and $h$ is the height of the prism.

The perimeter of the triangular base is $P = 9 + 9 + 9 = 27$ cm.

The height of the prism is $h = 12$ cm.

So, the lateral area is:

$L = 27 \times 12 = 324$ cm$^2$.

The surface area of a prism is given by the formula:

$S = L + 2B$

where $L$ is the lateral area and $B$ is the area of the base.

The area of the triangular base is given by:

$B = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 9 \times 7.8 = \frac{70.2}{2} = 35.1$ cm$^2$.

The surface area is:

$S = 324 + 2(35.1) = 324 + 70.2 = 394.2$ cm$^2$.

3. Final Answer

$L = 324.0$ cm$^2$

$S = 394.2$ cm$^2$