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The diagram provided shows a right triangle \( \triangle ABC \) with the following details:

- \( AB = 13 \)
- \( AC = 12 \)
- \( BM = 8 \)
- \( \angle A \) is a right angle

We can deduce that \( \triangle ABC \) is a right triangle with \( \angle A \) as the right angle. Let's solve for the missing side \( BC \) and verify the length of \( BM \).

### Step-by-Step Solution:

1. **Identify the missing side \( BC \)**:
   Since \( \triangle ABC \) is a right triangle, we can use the Pythagorean theorem:
   \[
   BC^2 = AB^2 + AC^2
   \]
   Substituting the given values:
   \[
   BC^2 = 13^2 + 12^2
   \]
   \[
   BC^2 = 169 + 144
   \]
   \[
   BC^2 = 313
   \]
   \[
   BC = \sqrt{313}
   \]

2. **Verification of BM**:
   The triangle \( \triangle ABM \) is also a right triangle (as BM is the height from B to the hypotenuse AC):
   \[
   BM^2 + AM^2 = AB^2
   \]
   Given \( BM = 8 \) and \( AB = 13 \), solve for \( AM \):
   \[
   AM^2 = AB^2 - BM^2
   \]
   \[
   AM^2 = 13^2 - 8^2
   \]
   \[
   AM^2 = 169 - 64
   \]
   \[
   AM^2 = 105
   \]
   \[
   AM = \sqrt{105}
   \]

3. **Height Verification**:
   If we interpret \( M \) as the midpoint of \( AC \) (though it's not explicitly stated), we need to verify if it matches the condition of midpoint:
   \[
   AM + MC = AC
   \]
   From our calculation:
   \[
   AM = \sqrt{105}
   \]
   \[
   MC = AC - AM = 12 - \sqrt{105}
   \]
   
   However, the segment MC does not directly match the previous conditions if \( M \) is not midpoint.

Thus, we have calculated:
- \( BC = \sqrt{313} \)
- \( AM = \sqrt{105} \)

Let me know if you need any further calculations or clarifications.

Would you like further details or explanations on any of the steps?

Here are some questions you can ask:
1. How to find the area of triangle \( ABC \)?
2. How to find the length of the median from \( B \) to \( AC \)?
3. How to verify if \( M \) is the midpoint of \( AC \)?
4. How to find the perimeter of triangle \( ABC \)?
5. How to find the coordinates of point \( M \) if \( A \) and \( B \) coordinates are known?

**Tip:** For right triangles, always start with the Pythagorean theorem to find missing side lengths.

Learn how to solve a right triangle ABC with sides AB = 13, AC = 12, and BM = 8. Detailed step-by-step solution using the Pythagorean theorem and properties of right triangles. Suitable for students in Grades 9-12.

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