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Use Pythagoras’ theorem to determine any unknown side lengths and find the total surface area of
these solids, correct to one decimal place

The image depicts a triangular prism with two right triangles at either end. We can see that the side lengths of the right triangle are 7, 8, and 10 units. Since one of these triangles forms the base of the prism, we'll calculate the surface area.

### Step 1: Verify the Triangle Using Pythagoras' Theorem
For a right triangle with legs \(a\), \(b\), and hypotenuse \(c\):

\[
c^2 = a^2 + b^2
\]

Here, we have:
- \(a = 6\)
- \(b = 8\)
- \(c = 10\)

Let's check if these values satisfy the Pythagorean theorem:

\[
10^2 = 7^2 + 8^2 \implies 100 = 49 + 64 = 113
\]

This indicates the triangle is not a right triangle, so the problem might be about a different triangle or there might be some details in the problem statement. However, let's proceed assuming these are the sides of the triangular faces.

### Step 2: Calculate the Surface Area of the Prism
To find the surface area, we need to calculate the areas of all the faces of the prism. The prism has two triangular faces and three rectangular faces.

#### 2.1: Area of the Triangular Faces
The area \(A_{\text{triangle}}\) of each triangular face is:

\[
A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \times 8 = 28 \text{ square units}
\]

There are two such triangular faces, so the total area of the triangular faces is:

\[
2 \times A_{\text{triangle}} = 2 \times 28 = 56 \text{ square units}
\]

#### 2.2: Area of the Rectangular Faces
The three rectangular faces are:
1. A face with dimensions 7 and 12.
2. A face with dimensions 8 and 12.
3. A face with dimensions 10 and 12.

The area of each rectangular face can be calculated as:

\[
A_{\text{rect1}} = 7 \times 10 = 70 \text{ square units}
\]

\[
A_{\text{rect2}} = 8 \times 10 = 80 \text{ square units}
\]

\[
A_{\text{rect3}} = 10 \times 10 = 100 \text{ square units}
\]

#### 2.3: Total Surface Area
Summing up the areas of the triangles and rectangles:

\[
\text{Total Surface Area} = A_{\text{triangles}} + A_{\text{rectangles}} = 56 + 70 + 80 + 100 = 306 \text{ square units}
\]

### Final Answer
The total surface area of the prism is approximately \(306\) square units.

---

Would you like more details or have any questions? Here are some related questions:

1. How do you calculate the surface area of different 3D shapes?
2. What is the significance of the Pythagorean Theorem in geometry?
3. How would the calculation change if the prism had different dimensions?
4. How do you find the volume of a triangular prism?
5. What are some applications of the Pythagorean Theorem in real-life problems?
6. How would you calculate the surface area if the triangle was not right-angled?
7. What are other methods to find the area of a triangle besides using the base and height?
8. How can you calculate the surface area of irregular shapes?

**Tip:** When dealing with composite shapes or solids, break them down into simpler components, solve for each part, and then sum the results for the total.

Learn how to use Pythagoras’ theorem to find unknown side lengths and calculate the total surface area of solids, correct to one decimal place. This problem involves applying Pythagoras’ theorem to verify side lengths and then using it to calculate the surface area of a triangular prism. Suitable for students in Grades 7-9.

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