https://r2.erweima.ai/mathbot/image/17259080039754zk1bfba.jpg

The problem states that both triangles \( \triangle ABC \) and \( \triangle ABD \) are right triangles. We are given the following:

- \( AC = 3 \)
- \( AB = 4 \)
- We need to find \( AD \).

### Step 1: Use the Pythagorean Theorem in \( \triangle ABC \)
Since \( \triangle ABC \) is a right triangle, we can use the Pythagorean theorem to find \( BC \). The theorem states:
\[
AB^2 = AC^2 + BC^2
\]
Substituting the known values of \( AB = 4 \) and \( AC = 3 \):
\[
4^2 = 3^2 + BC^2
\]
\[
16 = 9 + BC^2
\]
\[
BC^2 = 7
\]
\[
BC = \sqrt{7}
\]

### Step 2: Analyze \( \triangle ABD \)
In \( \triangle ABD \), since it is also a right triangle, we know that \( AD \) is the leg of this triangle, and \( BD \) is part of the hypotenuse of \( \triangle ABC \). From the image, it seems that \( D \) lies on \( BC \), meaning that \( BD + DC = BC \).

However, we do not have direct information about the position of \( D \), so further assumptions or information might be required to determine the length of \( AD \). But based on the given figure, the next logical step involves understanding more about the relationship between \( D \), \( C \), and the length of \( BD \).

Do you want to explore any additional details or information to clarify this?

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### Follow-up questions:
1. Can you provide more information about point \( D \) and its position on \( BC \)?
2. Is \( BD \) given or calculable using other relationships?
3. Would you like help using trigonometric ratios to solve for \( AD \)?
4. Do you need further clarification on the application of the Pythagorean theorem in this problem?
5. Would it help to explore different methods to find \( AD \), such as using geometric properties?

**Tip**: When solving right triangle problems, always check if additional relationships (like angles or proportions) between the triangles can simplify the solution.

Supóngase que ΔABC y ΔABD son triángulos rectángulos. Si AC = 3 y AB = 4, encuéntrese AD.

Learn how to solve for AD in the right triangles ΔABC and ΔABD where AC = 3, AB = 4. This problem uses the Pythagorean theorem to calculate BC and explores relationships between the triangles. Suitable for students in Grades 9-11.

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