https://r2.erweima.ai/mathbot/image/1725569996854prl5lwvg.jpg

To solve for \(\tan(\alpha)\) in the given triangle, let's analyze the right triangle \(ABC\) where \(\alpha\) is the angle at vertex \(A\). 

From the image, we can observe the following:

- The side opposite to angle \(\alpha\) is \(BC = 21\),
- The side adjacent to angle \(\alpha\) is \(AB = 29\),
- The hypotenuse is \(AC = 20\).

For a right triangle, the tangent of an angle \(\alpha\) is given by:

\[
\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AB}
\]

Substituting the values from the triangle:

\[
\tan(\alpha) = \frac{21}{29}
\]

Now, the question asks for the value of \(\tan(\alpha)\), and from the choices, the correct answer is:

\[
\boxed{\frac{21}{29}}
\]

Would you like further details or explanations? Here are 5 related questions you might find interesting:

1. How can we calculate \(\sin(\alpha)\) and \(\cos(\alpha)\) for this triangle?
2. How do we find the angle \(\alpha\) using \(\tan^{-1}\) on a calculator?
3. What is the relationship between the sides of a right triangle and the Pythagorean theorem?
4. How can the values of trigonometric functions help solve real-life problems involving triangles?
5. Why does the tangent function specifically involve the opposite and adjacent sides?

**Tip**: The tangent function is particularly useful when you know the lengths of the opposite and adjacent sides of a right triangle.

Learn how to calculate the tangent of an angle in a right triangle. This problem involves basic trigonometry, using the formula tan(α) = opposite / adjacent. Given side lengths of 21 and 29, we find tan(α) = 21/29. Perfect for students in Grades 9-12.

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