using the given diagram, solve for $x$.\n$x = \\square$ (simplify your answer.)

Explanation:

Step1: Identify the geometric theorem

The diagram shows a triangle with an angle bisector, so we apply the Triangle Angle Bisector Theorem.

Step2: Set up the proportionality ratio

The theorem states that the bisector divides the opposite side into segments proportional to the adjacent sides. $$\frac{8}{2} = \frac{20}{x}$$

Step3: Solve for x

Cross-multiply to solve the linear equation. $$8x = 2 \cdot 20$$ $$8x = 40$$ $$x = \frac{40}{8}$$ $$x = 5$$

Answer:

x = 5