Geometry

Question Analysis

The main focus is using the Triangle Mid - Segments Theorem to establish relationships between the lengths of line segments , , and . We'll set up equations based on the theorem and the mid - point properties in  to solve for the variables  and , and then find the lengths of  and .

Key Concept Explanation

The Triangle Mid - Segments Theorem states that a mid - segment of a triangle (a line segment connecting the midpoints of two sides of a triangle) is parallel to the third side of the triangle and half its length. In ,  is a mid - segment, so . Also, since  is the mid - point of , the segments on  (i.e.,  and ) are equal.

Step-by-Step Solution

1. Solve for  using the mid - segment relationship:

Given , with  and , we set up the equation .

Multiply both sides by 2 to get rid of the fraction: .

Expand the left side: .

Subtract  from both sides: , which simplifies to .

Add 8 to both sides: , and then divide by 4 to obtain

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