Triangle A B C, but angle A is bisected by line segment A D, creating two new triangles, triangle A C D and triangle A B D. Point D is on Side B C. Side A C is five point eight units. Side C D is two point five units. Side A B is six point two units. What is the length of 
 
\[\overline{BD}\]?
Round to one decimal place.

To find the length of \(\overline{BD}\), we can use the Angle Bisector Theorem. This theorem states that if a line segment bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the lengths of the other two sides of the triangle.

Given:
- \(AC = 5.8\) units
- \(CD = 2.5\) units
- \(AB = 6.2\) units

Let \(BD = x\) and \(DC = 2.5\).

According to the Angle Bisector Theorem:
\[
\frac{BD}{DC} = \frac{AB}{AC}
\]

Substituting the known values:
\[
\frac{x}{2.5} = \frac{6.2}{5.8}
\]

First, we calculate \(\frac{6.2}{5.8}\):
\[
\frac{6.2}{5.8} \approx 1.06896551724
\]

Now we set up the equation:
\[
\frac{x}{2.5} = 1.06896551724
\]

Solving for \(x\):
\[
x = 2.5 \times 1.06896551724
\]
\[
x \approx 2.6724137931
\]

Rounding to one decimal place:
\[
BD \approx 2.7 \text{ units}
\]

So, the length of \(\overline{BD}\) is approximately \(2.7\) units.

Would you like any further details or have any questions?

Here are 8 related questions that might interest you:
1. How is the Angle Bisector Theorem derived?
2. Can the Angle Bisector Theorem be used in non-Euclidean geometries?
3. What are the applications of the Angle Bisector Theorem in real-life problems?
4. How do you find the coordinates of point D if the coordinates of points A, B, and C are given?
5. How does the Angle Bisector Theorem relate to the concept of similar triangles?
6. What if angle bisector \(AD\) does not intersect at \(D\) on \(BC\)? What are the implications?
7. How can we use the Angle Bisector Theorem in trigonometric problems?
8. Can you prove the Angle Bisector Theorem using vectors?

**Tip:** When dealing with geometric problems, drawing a clear diagram can significantly help in understanding the relationships between different parts of the figure.

Learn how to calculate the length of segment BD in triangle ABC using the Angle Bisector Theorem. Given side lengths AC = 5.8 units, CD = 2.5 units, and AB = 6.2 units, find the length of BD rounded to one decimal place.

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