https://file.aiquickdraw.com/mathbot/file/172827317244259oxfvsr.jpg

Let's analyze the problem step by step, based on the image provided.

### (a) Find \( a \):
In this problem, the triangle that includes \( a \), the slant height of the side face, forms a right triangle. From the given figure, we know:
- The height of the pyramid is \( 4 \, \text{m} \),
- The distance from the center to the base of the square is \( 3 \, \text{m} \) (half of the total base length which is \( 6 \, \text{m} \)).

So, we apply the **Pythagorean theorem** to find \( a \), where the triangle formed has legs of \( 3 \, \text{m} \) and \( 4 \, \text{m} \).

\[
a^2 = 3^2 + 4^2
\]
\[
a^2 = 9 + 16
\]
\[
a^2 = 25
\]
\[
a = \sqrt{25} = 5 \, \text{m}
\]

Thus, \( a = 5 \, \text{m} \).

### (b) Find \( b \), the length of one string of lights:
The string of lights forms the hypotenuse of another right triangle with \( a \) and \( 3 \, \text{m} \) as the legs. We already found that \( a = 5 \, \text{m} \).

Now, applying the **Pythagorean theorem** again to find \( b \):

\[
b^2 = a^2 + 3^2
\]
\[
b^2 = 5^2 + 3^2
\]
\[
b^2 = 25 + 9
\]
\[
b^2 = 34
\]
\[
b = \sqrt{34} \approx 5.8 \, \text{m}
\]

Thus, \( b \approx 5.8 \, \text{m} \).

### Final answers:
- \( a = 5 \, \text{m} \)
- \( b \approx 5.8 \, \text{m} \)

Would you like more details on any part of this explanation?

#### Related Questions:
1. What happens if the height of the pyramid changes?
2. How would you calculate the surface area of the pyramid?
3. How is the Pythagorean theorem used in 3D geometry?
4. How would you approach finding the total length of lights needed for all edges?
5. Can you calculate the volume of the pyramid based on the given dimensions?

#### Tip:
When solving 3D problems, always identify all the right triangles involved—breaking the problem down into smaller, manageable parts using the Pythagorean theorem helps simplify complex geometry!

An architecture firm is designing a pavilion in the shape of a square pyramid. They plan to add a string of lights of length b (in meters) to each of the edges that meet at the top. In the figure, the edge with length b shows the placement of one string of lights. (The figure is not drawn to scale.)

(a) Find a.

(b) Use your answer to part (a) to find b, the length of one string of lights. Round your answer to the nearest tenth of a meter.

Learn how to calculate the slant height and the string length of lights for a square pyramid pavilion design using the Pythagorean Theorem. This problem involves applying 3D geometry concepts and using the Pythagorean theorem in multiple steps. Suitable for students in Grades 9-12.

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