Geometry

Question Analysis

The main focus is applying the perpendicular bisector theorem. Since  is the perpendicular bisector of  (marked by the right - angle at  and equal - length segments on ), we need to find  and  by first determining the value of  from the equal - length relationship of  and .

Key Concept Explanation

The perpendicular bisector theorem states that if a point lies on the perpendicular bisector of a line segment, it is equidistant from the endpoints of that line segment. Here, for line segment  with perpendicular bisector , we have .

Step-by-Step Solution

1. Set up the equation:

Because  according to the perpendicular bisector theorem, we get .

2. Solve for :

Subtract  from both sides: , which simplifies to .

Add 6 to both sides:

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