To solve the given problem, let's use the properties of similar triangles and the Pythagorean theorem.

Step 1: Identify Similar Triangles

In right triangle $\triangle ABC$, the altitude $BD$ divides it into two smaller right triangles: $\triangle ABD$ and $\triangle CBD$. These triangles are similar to each other and to $\triangle ABC$ due to the AA (Angle-Angle) similarity criterion.

Thus, we have: $\triangle ABC \sim \triangle ABD \sim \triangle CBD$

Step 2: Use the Similarity Ratios

The property of similar triangles tells us: $\frac{AD}{AB} = \frac{AB}{AC} \quad \text{(for } \triangle ABD \sim \triangle ABC\text{)}.$

Let $AB = x$. Then substitute the given values $AD = 1$ and $AC = 4$ into the similarity equation: $\frac{1}{x} = \frac{x}{4}.$

Step 3: Solve for $x$

Cross-multiply: $x^2 = 4.$ Taking the positive square root (since length is positive): $x = 2.$

Final Answer:

The length of $AB$ is: $\boxed{2}.$

Would you like further clarification or additional details? Here are some related questions for practice:

1. How can you confirm the similarity of triangles using angles?
2. What is the relationship between the geometric mean and the altitude in a right triangle?
3. How do you derive the equation for $AB$ using the Pythagorean theorem instead of similarity?
4. What are the side ratios of the smaller triangles $\triangle ABD$ and $\triangle CBD$?
5. Can you calculate the length of $BD$ using this setup?

Tip: In problems involving right triangles and altitudes, always check for similar triangles and use their properties for efficient solutions.