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In the given triangle \( \triangle ABC \), the following information is provided:

- \( \ell(AB) = 6 \)
- \( \ell(BC) = 7 \)
- \( \ell(AC) = 8 \)
- \( AD \) is perpendicular to \( BC \)
- \( AE \) is an angle bisector of \( \angle A \)
- \( AF \) is a median of \( \triangle ABC \)

### Key Definitions:
1. **Perpendicular line (AD)**: This means \( AD \perp BC \), so \( \triangle ABD \) and \( \triangle ADC \) are right triangles. 
2. **Angle bisector (AE)**: Since \( AE \) bisects \( \angle A \), it divides \( \angle BAC \) into two equal angles. Also, by the Angle Bisector Theorem, the sides opposite these angles have a specific ratio.
3. **Median (AF)**: \( AF \) is the median, so \( F \) is the midpoint of \( BC \). Therefore, \( BF = FC = \frac{7}{2} = 3.5 \).

### Tasks:
From this setup, you can explore properties related to:
- **Area of the triangle**: Using the given sides and the perpendicular, you can find the height and apply the area formula for a triangle.
- **Right triangle relationships**: Apply the Pythagorean theorem in \( \triangle ABD \) and \( \triangle ADC \).
- **Angle Bisector Theorem**: This can help in finding relations between the sides of the triangle.
- **Coordinate geometry**: Assigning coordinates to the vertices of the triangle might be useful for further calculations.

Would you like to explore a specific aspect or solve for something specific? Here are some questions to guide further exploration:

1. What is the length of the height \( AD \) using the Pythagorean theorem?
2. Can you calculate the area of \( \triangle ABC \) using \( AD \) as the height?
3. What is the length of \( AE \) using the Angle Bisector Theorem?
4. How can you use the median \( AF \) to explore the balance between sides \( AB \) and \( AC \)?
5. What are the internal angles of \( \triangle ABC \), and how do the bisectors divide them?

### Tip:
Remember that the properties of perpendiculars, angle bisectors, and medians divide the triangle in various meaningful ways that can help simplify calculations.

In the triangle shown in the figure, l(AB) = 6, l(BC) = 7, and l(AC) = 8. Also, AD is a perpendicular to BC, AE is an angle bisector of ∠A, and AF is a median of ΔABC.

Explore the properties of triangle ABC with side lengths AB = 6, BC = 7, and AC = 8. Learn how to calculate the length of the height, the median, and the angle bisector using geometric principles such as the Pythagorean theorem, angle bisector theorem, and properties of medians. Suitable for students in Grades 9-12.

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