Mid-Segment of a Triangle
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Definition:

The mid-segment of a triangle (also called a midline) is a segment joining the midpoints of two sides of a triangle.

 

   

Properties:

1.

 

 The mid-segment of a triangle joins the midpoints of two sides of a triangle such that it is parallel to the third side of the triangle.
 
 

2.

 

The mid-segment of a triangle joins the midpoints of two sides of a triangle such that its length is half the length of the third side of the triangle.

 

Examples:

1. Given DE is the length of the mid-segment.  Find AB.   Solution:
The mid-segment is half of the third side.
7 is half of 14.
AB = 14.

 

     

 
2. Given DE, DF, and FE are the lengths of mid-segments.  Find the perimeter of triangle ABC.   Solution:
The mid-segment is half of the third side.
6 is half of 12 so AC = 12
7 is half of 14 so CB = 14
8 is half of 16 so AB = 16
The perimeter of the large triangle ABC is:
 12 + 14 + 16 = 42.
       

 

3.

 
Statements Reasons
1. 1. Given
2. 2. The mid-segment of a triangle is parallel to the third side of the triangle.
3. 3. If two parallel lines are cut by a transversal, the corresponding angles are congruent.
4. 4. Transitive property.
5. 5. The midpoint of a segment divides the segment into two congruent segments.
6. 6. AAS.  If 2 angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.