Question

segment \\( \overline{bd} \\) bisects \\( \angle abc \\). solve for \\( x \\). round to the nearest tenth, if necessary. (image not necessarily to scale.)

Explanation:

Step1: Apply Angle Bisector Theorem

The Angle Bisector Theorem states that if a bisector of an angle of a triangle divides the opposite side into segments proportional to the adjacent sides. So, $\frac{AB}{BC} = \frac{AD}{DC}$. Let $AB = x$, $BC = 10$, $AD = 6$, $DC = 5$. Then we have $\frac{x}{10} = \frac{6}{5}$.

Step2: Solve for \( x \)

Cross - multiply the proportion: $5x = 10\times6$.
Simplify the right - hand side: $5x = 60$.
Divide both sides by 5: $x=\frac{60}{5}=12$.

Answer:

\( x = 12 \)