Practice Makes Perfect: Solve for variables in right triangles using trigonometric relationships.

Problem Overview:

The task involves solving for the values of variables in various geometric figures, specifically right triangles and 30-60-90 triangles. The solutions should be expressed as integers or in simplest radical form if they are not integers.

Key Properties Used:

1. 30-60-90 Triangle Properties:
- In a 30-60-90 triangle, the sides are in the ratio:
\( \text{Short leg} : \text{Long leg} : \text{Hypotenuse} = x : x\sqrt{3} : 2x \).
- The side opposite the 30° angle is the shortest side.
- The side opposite the 60° angle is \( x\sqrt{3} \).
- The hypotenuse is twice the length of the shortest side.

2. Pythagorean Theorem:
- For any right triangle with legs \( a \) and \( b \), and hypotenuse \( c \):
\( a^2 + b^2 = c^2 \).

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Solutions for Each Figure:

#### Figure 1:
- Given: A 30-60-90 triangle with the hypotenuse \( 8 \).
- Using the 30-60-90 triangle properties:
- Hypotenuse = \( 2x \), so \( 2x = 8 \) → \( x = 4 \).
- Short leg = \( x = 4 \).
- Long leg = \( x\sqrt{3} = 4\sqrt{3} \).
- Therefore, \( t = 4 \).

#### Figure 2:
- Given: A 30-60-90 triangle with the short leg \( t \) and hypotenuse \( 8 \).
- Using the 30-60-90 triangle properties:
- Hypotenuse = \( 2x \), so \( 2x = 8 \) → \( x = 4 \).
- Short leg = \( x = 4 \).
- Therefore, \( t = 4 \).

#### Figure 3:
- Given: A 30-60-90 triangle with the long leg \( t \) and hypotenuse \( 8 \).
- Using the 30-60-90 triangle properties:
- Hypotenuse = \( 2x \), so \( 2x = 8 \) → \( x = 4 \).
- Long leg = \( x\sqrt{3} = 4\sqrt{3} \).
- Therefore, \( t = 4\sqrt{3} \).

#### Figure 4:
- Given: A 30-60-90 triangle with the short leg \( t \) and long leg \( \sqrt{3} \).
- Using the 30-60-90 triangle properties:
- Long leg = \( x\sqrt{3} \), so \( x\sqrt{3} = \sqrt{3} \) → \( x = 1 \).
- Short leg = \( x = 1 \).
- Therefore, \( t = 1 \).

#### Figure 5:
- Given: A 30-60-90 triangle with the short leg \( t \) and hypotenuse \( m \).
- Using the 30-60-90 triangle properties:
- Hypotenuse = \( 2x \), so \( m = 2x \).
- Short leg = \( x = t \).
- Therefore, \( m = 2t \).

#### Figure 6:
- Given: A 30-60-90 triangle with the short leg \( t \) and long leg \( \sqrt{3} \).
- Using the 30-60-90 triangle properties:
- Long leg = \( x\sqrt{3} \), so \( x\sqrt{3} = \sqrt{3} \) → \( x = 1 \).
- Short leg = \( x = 1 \).
- Therefore, \( t = 1 \).

#### Figure 7:
- Given: A 30-60-90 triangle with the hypotenuse \( 11\sqrt{2} \) and one leg \( t \).
- Using the 30-60-90 triangle properties:
- Hypotenuse = \( 2x \), so \( 2x = 11\sqrt{2} \) → \( x = \frac{11\sqrt{2}}{2} \).
- Short leg = \( x = \frac{11\sqrt{2}}{2} \).
- Long leg = \( x\sqrt{3} = \frac{11\sqrt{2}}{2} \cdot \sqrt{3} = \frac{11\sqrt{6}}{2} \).
- Since \( t \) is the short leg, \( t = \frac{11\sqrt{2}}{2} \).

#### Figure 8:
- Given: A 30-60-90 triangle with the short leg \( t \) and hypotenuse \( m \).
- Using the 30-60-90 triangle properties:
- Hypotenuse = \( 2x \), so \( m = 2x \).
- Short leg = \( x = t \).
- Therefore, \( m = 2t \).

#### Figure 9:
- Given: A 30-60-90 triangle with the short leg \( t \) and hypotenuse \( 10 \).
- Using the 30-60-90 triangle properties:
- Hypotenuse = \( 2x \), so \( 2x = 10 \) → \( x = 5 \).
- Short leg = \( x = 5 \).
- Therefore, \( t = 5 \).

#### Figure 10:
- Given: A 30-60-90 triangle with the short leg \( t \) and hypotenuse \( 12 \).
- Using the 30-60-90 triangle properties:
- Hypotenuse = \( 2x \), so \( 2x = 12 \) → \( x = 6 \).
- Short leg = \( x = 6 \).
- Therefore, \( t = 6 \).

#### Figure 11:
- Given: A 30-60-90 triangle with the short leg \( t \) and hypotenuse \( 6\sqrt{3} \).
- Using the 30-60-90 triangle properties:
- Hypotenuse = \( 2x \), so \( 2x = 6\sqrt{3} \) → \( x = 3\sqrt{3} \).
- Short leg = \( x = 3\sqrt{3} \).
- Therefore, \( t = 3\sqrt{3} \).

#### Figure 12:
- Given: A 30-60-90 triangle with the short leg \( t \) and hypotenuse \( 6\sqrt{3} \).
- Using the 30-60-90 triangle properties:
- Hypotenuse = \( 2x \), so \( 2x = 6\sqrt{3} \) → \( x = 3\sqrt{3} \).
- Short leg = \( x = 3\sqrt{3} \).
- Therefore, \( t = 3\sqrt{3} \).

#### Figure 13:
- Given: A 30-60-90 triangle with the short leg \( t \) and hypotenuse \( 7 + \sqrt{3} \).
- Using the 30-60-90 triangle properties:
- Hypotenuse = \( 2x \), so \( 2x = 7 + \sqrt{3} \) → \( x = \frac{7 + \sqrt{3}}{2} \).
- Short leg = \( x = \frac{7 + \sqrt{3}}{2} \).
- Therefore, \( t = \frac{7 + \sqrt{3}}{2} \).

#### Figure 14:
- Given: A 30-60-90 triangle with the short leg \( t \) and hypotenuse \( 8 + 2\sqrt{3} \).
- Using the 30-60-90 triangle properties:
- Hypotenuse = \( 2x \), so \( 2x = 8 + 2\sqrt{3} \) → \( x = \frac{8 + 2\sqrt{3}}{2} = 4 + \sqrt{3} \).
- Short leg = \( x = 4 + \sqrt{3} \).
- Therefore, \( t = 4 + \sqrt{3} \).

#### Figure 15:
- Given: A 30-60-90 triangle with the short leg \( t \) and hypotenuse \( 10 \).
- Using the 30-60-90 triangle properties:
- Hypotenuse = \( 2x \), so \( 2x = 10 \) → \( x = 5 \).
- Short leg = \( x = 5 \).
- Therefore, \( t = 5 \).

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Final Answers:

\[
\boxed{
\begin{aligned}
1. & \quad t = 4 \\
2. & \quad t = 4 \\
3. & \quad t = 4\sqrt{3} \\
4. & \quad t = 1 \\
5. & \quad m = 2t \\
6. & \quad t = 1 \\
7. & \quad t = \frac{11\sqrt{2}}{2} \\
8. & \quad m = 2t \\
9. & \quad t = 5 \\
10. & \quad t = 6 \\
11. & \quad t = 3\sqrt{3} \\
12. & \quad t = 3\sqrt{3} \\
13. & \quad t = \frac{7 + \sqrt{3}}{2} \\
14. & \quad t = 4 + \sqrt{3} \\
15. & \quad t = 5 \\
\end{aligned}
}
\]