We are given a triangle with a line segment that bisects an angle of the triangle. We are given the lengths of some sides and segments and are asked to find the length of the side labeled $x$. The lengths given are 4.7, 7.7 and 15.4. The angle bisector divides the side with length 15.4.

1. Problem Description

We are given a triangle with a line segment that bisects an angle of the triangle. We are given the lengths of some sides and segments and are asked to find the length of the side labeled $x$. The lengths given are 4.7, 7.7 and 15.

4. The angle bisector divides the side with length 15.

4.

2. Solution Steps

The Angle Bisector Theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the lengths of the other two sides. In our case, the angle bisector divides the side of length 15.4 into segments of lengths 7.7 and 4.

7. The sides of the triangle are $x$, 15.4 and the remaining side. Therefore, we can write the proportion:

$\frac{x}{4.7} = \frac{15.4}{7.7}$

Cross-multiplying, we get:

$7.7x = 4.7 \times 15.4$

$7.7x = 72.38$

$x = \frac{72.38}{7.7}$

$x \approx 9.4$

3. Final Answer

9. 4