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Find the length of AB

A. 66.35
B. 48.21
C. 28.34
D. 53.68

To find the length of \( AB \) in the given right triangle \( \triangle ABC \):

1. **Identify the known elements:**
   - \( \angle ACB = 90^\circ \)
   - \( \angle BAC = 54^\circ \)
   - \( AC = 39 \) (opposite side to the given angle)

2. **Determine \( \angle ABC \):**
   \[
   \angle ABC = 90^\circ - 54^\circ = 36^\circ
   \]

3. **Use the sine function to find \( AB \):**
   - In right triangles, \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\).

   For \(\angle BAC = 54^\circ\):
   \[
   \sin(54^\circ) = \frac{39}{AB}
   \]

   Rearrange to solve for \( AB \):
   \[
   AB = \frac{39}{\sin(54^\circ)}
   \]

4. **Calculate \( AB \):**
   \[
   \sin(54^\circ) \approx 0.8090
   \]
   \[
   AB = \frac{39}{0.8090} \approx 48.21
   \]

So, the length of \( AB \) is approximately \( 48.21 \).

Therefore, the correct answer is:
\[ \boxed{48.21} \]

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How would you find the length of \( BC \)?
2. What trigonometric ratio would you use to find \( BC \)?
3. Can you verify the length of \( AB \) using the cosine function?
4. How do you find the length of \( AB \) using the Pythagorean theorem after finding \( BC \)?
5. What is the area of \( \triangle ABC \)?
6. How would you find the perimeter of \( \triangle ABC \)?
7. Can you find the length of \( AB \) using the tangent function and the other angle?
8. What would the hypotenuse be if \( \angle BAC \) were \( 30^\circ \) and \( AC \) remained the same?

**Tip:** When solving for lengths in right triangles, always remember the basic trigonometric ratios (sine, cosine, tangent) and their respective definitions.

Learn how to find the length of AB in a right triangle with the given angle and side using trigonometric functions. This problem involves applying the sine function to calculate the opposite side in a right triangle. Suitable for high school students learning trigonometry.

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