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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.

To solve this problem, we will use the Pythagorean theorem to determine any unknown side lengths of the triangular faces, and then we'll calculate the total surface area of the solid.

### Step 1: Identify the Shape and Given Dimensions

The image shows a triangular pyramid (tetrahedron) where:
- One right triangle has sides 7, 8, and 10 units.
- The other triangle also appears to be a right triangle, with a side length of 7 units and a height of 8 units.

### Step 2: Calculate the Unknown Side Lengths

#### First Triangle (Right Triangle):
The first triangle is a right triangle with sides:
- \(a = 7\)
- \(b = 8\)
- \(c = 10\) (the hypotenuse)

This triangle is already defined, so no further calculation is needed here.

#### Second Triangle:
This second triangle is also a right triangle:
- One leg is 7 units
- The other leg is 8 units

We need to calculate the hypotenuse (\(c\)) of this triangle using the Pythagorean theorem:

\[
c = \sqrt{7^2 + 8^2} = \sqrt{49 + 64} = \sqrt{113} \approx 10.6 \text{ units}
\]

### Step 3: Calculate the Surface Area

The tetrahedron consists of four triangular faces:
- Two triangles with sides \(7, 8, 10\) units.
- Two triangles with sides \(7, 8, 10.6\) units.

#### 3.1: Area of Each Triangle

1. **Triangle with sides 7, 8, and 10:**
   Since it's a right triangle, the area is:
   \[
   \text{Area} = \frac{1}{2} \times 7 \times 8 = 28 \text{ square units}
   \]

2. **Triangle with sides 7, 8, and 10.6:**
   Similarly, since this is also a right triangle:
   \[
   \text{Area} = \frac{1}{2} \times 7 \times 8 = 28 \text{ square units}
   \]

#### 3.2: Total Surface Area

The tetrahedron has four triangular faces, so the total surface area is the sum of the areas of these four triangles:

\[
\text{Total Surface Area} = 28 + 28 + 28 + 28 = 112 \text{ square units}
\]

### Final Answer
The total surface area of the solid is **112 square units**, rounded to one decimal place.

---

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

1. How do you calculate the surface area of other 3D shapes like pyramids and cylinders?
2. How would you approach this problem if the triangles were not right triangles?
3. What is the difference between a prism and a pyramid in terms of surface area calculation?
4. How can you find the volume of this tetrahedron?
5. How can the Pythagorean theorem be applied to find distances in 3D space?
6. What if the given sides were in different units—how would you convert them?
7. How do you calculate the area of an irregular polygon?
8. What are some real-life applications of calculating surface area?

**Tip:** When working with 3D shapes, always ensure you correctly identify the faces and their corresponding shapes to accurately calculate the surface area.

Learn how to apply Pythagoras’ theorem to determine unknown side lengths and find the total surface area of a solid with triangular faces. This problem involves calculating the areas of right triangles and applying the Pythagorean theorem in a geometric context. Suitable for Grades 7-9 students.

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