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, $\frac{AD}{DC} = \frac{AB}{BC}$. Here, $AD = 5$, $DC = 6$, $BC = 10$, and $AB = x$.

Step2: Substitute values into the theorem

Substitute the known values into the proportion: $\frac{5}{6} = \frac{x}{10}$.

Step3: Solve for \( x \)

Cross - multiply to get $6x = 5\times10$. Then $6x = 50$. Divide both sides by 6: $x=\frac{50}{6}\approx8.3$ (rounded to the nearest tenth).

Answer:

\( x\approx8.3 \)