Geometry

Question Analysis

The main focus is using the Triangle Mid - Segments Theorem to find the lengths of  and . We'll set up equations based on the mid - segment relationship between  and , and the mid - point relationship on side . Then solve for  and  to determine the lengths.

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. Here,  is a mid - segment of . Also, since  is the mid - point of , .

Step-by-Step Solution

1. Find the value of  using the mid - segment relationship:

Since  is a mid - segment of , we have .

Given  and , we set up the equation .

Multiply both sides by 2 to get .

Expand: .

Subtract  from both sides: , so .

Add 6 to both sides: , then .

2. Calculate the length of :

Substitute