Geometry

Question Analysis

The main focus is using the property of the perpendicular bisector  of  to find the lengths of  and . We'll first find the value of  based on the equal - length relationship from the perpendicular bisector theorem and then calculate the required lengths.

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, since  is the perpendicular bisector of  (shown by the right - angle at  and equal - length markings on  and ), 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 3 to both sides:

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