Question

find the value of x rounded to the nearest tenth.
12.0
8.0
12.5
15.5

Explanation:

Step1: Apply the angle - bisector theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the other two sides. Let the triangle have sides \(a = 10\), \(b = 8\), \(c = 10\), and the segment of the side opposite the bisected - angle be \(x\) and the other part be \(y\). According to the angle - bisector theorem, \(\frac{10}{8}=\frac{x}{y}\). Also, assume the whole side length is \(s=x + y\). In this case, we can use another approach. Consider the two similar triangles formed by the angle - bisector. Let the two triangles be \(\triangle ABC\) and \(\triangle ABD\) (where \(AD\) is the angle - bisector).
We know that if two triangles are similar, the ratios of their corresponding sides are equal. Let's use the property of similar triangles. The two triangles formed by the angle - bisector are similar. Let the large triangle have sides \(10\) and \(10 + 8=18\), and the small triangle has sides \(10\) and \(x\).
Since the two triangles are similar, we have the proportion \(\frac{x}{10}=\frac{10 + 8}{10}\).

Step2: Solve the proportion for \(x\)

Cross - multiply the proportion \(\frac{x}{10}=\frac{18}{10}\). We get \(x=\frac{10\times18}{10}=12.5\).

Answer:

12.5