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Let's analyze the given problem.

We are dealing with a right triangular prism, and we need to find:
1. The length of **a** (the diagonal of the right triangle base of the prism) using the dimensions provided: \(12 \, \text{in}\) and \(9 \, \text{in}\).
2. The length of **b** (the diagonal of the prism extending between two non-adjacent corners), once we know **a** and another dimension, \(7 \, \text{in}\).

### Part (a): Finding **a**

Since **a** is the hypotenuse of the right triangle with legs \(12 \, \text{in}\) and \(9 \, \text{in}\), we can use the Pythagorean Theorem:
\[
a^2 = 12^2 + 9^2
\]
Let me calculate that.

### Part (b): Finding **b**

Once we have **a**, we know that **b** forms another right triangle where **a** and the side \(7 \, \text{in}\) are the legs, and **b** is the hypotenuse:
\[
b^2 = a^2 + 7^2
\]
I'll calculate both of these values.### Part (a): 
The length of **a** is \( 15.0 \, \text{inches} \).

### Part (b): 
The length of **b** (the length of the tape) is approximately \( 16.6 \, \text{inches} \), rounded to the nearest tenth.

Would you like further clarification on any step? Here are 5 related questions for deeper exploration:

1. How is the Pythagorean Theorem applied in 3D geometry problems like this one?
2. What would happen if one of the dimensions was different? How would it affect **a** and **b**?
3. Can you describe how the length of **b** changes as the height of the prism increases?
4. What is the significance of understanding diagonal lengths in real-world objects like boxes or prisms?
5. How can trigonometry extend these concepts to angles within the prism?

**Tip**: Always double-check units in geometric problems to ensure all dimensions are consistent.

A gift box has the shape of a right triangular prism. A piece of tape of length b (in inches) is placed from one corner to the opposite corner, as shown in the figure. Find (a) the length of a and (b) the length of b, rounded to the nearest tenth of an inch.

Learn how to find the lengths of diagonals in a right triangular prism using the Pythagorean Theorem. This problem involves calculating the hypotenuses of right triangles with given side lengths. Suitable for high school students in Grades 9-12.

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