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, $\frac{BC}{AB} = \frac{CD}{DA}$. Here, $BC = 12$, $AB = 14$, $CD = 7$, and $DA = x$.

Step2: Substitute values into the proportion

Substitute the known values into the proportion: $\frac{12}{14} = \frac{7}{x}$.

Step3: Cross - multiply to solve for x

Cross - multiplying gives $12x = 14\times7$.

Step4: Calculate the right - hand side

Calculate $14\times7 = 98$. So, $12x = 98$.

Step5: Solve for x

Divide both sides by 12: $x=\frac{98}{12}\approx8.2$.

Answer:

$x\approx8.2$