Triangle ABC, ac is 12, ab is 10, bc is 8 . M is on BC and angle BAM equals angle cam. So what is bm

Given triangle \( \triangle ABC \) with sides:
- \( AC = 12 \)
- \( AB = 10 \)
- \( BC = 8 \)

Point \( M \) is on \( BC \) such that \( \angle BAM = \angle CAM \). This implies \( M \) is the point where the angle bisector of \( \angle BAC \) intersects \( BC \).

To find \( BM \), we can use the Angle Bisector Theorem, which states that the angle bisector divides the opposite side in the ratio of the other two sides.

### Step-by-Step Solution:

1. **Using the Angle Bisector Theorem**:
   The Angle Bisector Theorem states:
   \[
   \frac{BM}{MC} = \frac{AB}{AC}
   \]
   Substituting the given values:
   \[
   \frac{BM}{MC} = \frac{10}{12} = \frac{5}{6}
   \]
   
2. **Setting up the proportion**:
   Let \( BM = x \) and \( MC = 8 - x \). Using the ratio from the Angle Bisector Theorem:
   \[
   \frac{x}{8 - x} = \frac{5}{6}
   \]

3. **Solving for \( x \)**:
   Cross-multiplying to solve for \( x \):
   \[
   6x = 5(8 - x)
   \]
   \[
   6x = 40 - 5x
   \]
   \[
   11x = 40
   \]
   \[
   x = \frac{40}{11}
   \]

Therefore, \( BM = \frac{40}{11} \approx 3.64 \).

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 \( CM \)?
3. How to verify the Angle Bisector Theorem?
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 angle bisectors, remember that they divide the opposite side into segments proportional to the other two sides.

Learn how to find BM in triangle ABC using the Angle Bisector Theorem. Given sides AC = 12, AB = 10, and BC = 8, and angle conditions, calculate BM step-by-step with detailed explanations. Suitable for Grades 9-12 students.

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