**Title: Solving for \( x \) in a Circle Geometry Problem****Problem Statement:**Solve for \( x \).**Diagram Description:**The diagram consists of a circle with a secant line intersecting it. The secant line creates two segments outside the circle: one segment is \( 8 \) units long and the other is \( 2 \) units long. The longer segment of the secant line appears to extend into the circle, creating an internal chord. The part of the secant line that is inside the circle is labeled \( x \), and the angle formed by the lines intersecting the circle is given as \( 66^\circ \).**Geometry Explanation:**In this type of problem involving a circle, the secant-tangent angle theorem might be useful. According to this theorem, the measure of an angle formed by two secant lines intersecting outside a circle is equal to half the difference of the intercepted arcs.Given:- External segments of the secant lines: \( 8 \) and \( 2 \) units.- Internal chord segment: \( x \).We can use the secant-segment theorem which states:\[ (external \ segment \ 1 + internal \ segment) \times external \ segment \ 1 = (external \ segment \ 2 + internal \ segment) \times external \ segment \ 2 \]Substituting the given values to solve for \( x \).Perfect understanding of circle theorems and their properties is crucial in solving such geometry problems. Detailed steps are needed to find the exact value of \( x \).

Name: **Title: Solving for \( x \) in a Circle Geometry Problem****Problem
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: **Title: Solving for \( x \) in a Circle Geometry Problem****Problem Statement:**Solve for \( x \).**Diagram Description:**The diagram consists of a circle with a secant line intersecting it. The secant line creates two segments outside the circle: