Question

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

Explanation:

Step1: Apply the Angle - Bisector Theorem

The Angle - Bisector Theorem states that if a segment bisects an angle of a triangle, then it divides the opposite side into segments proportional to the adjacent sides. In \(\triangle ABC\) with angle bisector \(BD\), we have \(\frac{AD}{DC}=\frac{AB}{BC}\). Here, \(AD = 10\), \(DC=11\), and \(AB = 13\). Let \(BC=x\). So, \(\frac{10}{11}=\frac{13}{x}\).

Step2: Cross - multiply to solve for \(x\)

Cross - multiplying the proportion \(\frac{10}{11}=\frac{13}{x}\) gives us \(10x=11\times13\). Then \(10x = 143\).

Step3: Isolate \(x\)

Divide both sides of the equation \(10x = 143\) by \(10\). So, \(x=\frac{143}{10}=14.3\).

Answer:

\(14.3\)