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

theorem
"Mid-Segment Theorem": The mid-segment of a triangle, which joins the midpoints of two sides of a triangle, is parallel to the third side of the triangle and half the length of that third side of the triangle.
midsegmentnotation
midseg1


Examples:
1. midsegment1
Given M, N midpoints.
MN = 12
Find DF.
ANSWER:
midsegment1ANS

 

2. midsegment2
Given D, E midpoints.
DE = 3x = 5
AB = 26
Find x.
ANSWER:
midsegment2ANS

 

3. midsegment3
Given right ΔRST.
G, N, J midpoints.
ST = 6; RS = 8
Find perimeter of ΔGNJ.

 

 

ANSWER:
midsegment3ANS




Proof of Mid-Segment Theorem - Using Coordinate Geometry

For this proof, the diagram has been positioned in the first quadrant with one side on the x-axis to keep the algebraic computations as simple as possible, without losing the general positioning of the triangle. Be aware that other positionings are also possible.
midsegmentCG
midsegmentCGproofGiven
Coordinate Geometry formulas needed for this proof:
Midpoint Formula: midpointformula
Distance Formula: distanceformula2

Proof:
midproofcoordinate

 

 

Proof of Mid-Segment Theorem - Using Similar Triangles

For this proof, we will prove ΔMFN is similar ΔDFE, by SAS for similar triangles, to obtain corresponding angles for parallel lines and establish a pair of proportional sides.
midsegmentSASdiagram
midsegmentSproofGiven
Statements
Reasons
1. SASproof1b
1. Given
2. SASProof2a
2. A mid-segment joins the midpoints of two sides of a triangle.
3. SASproof3
3. Midpoint of a segment divides a segment into 2 congruent segments.
4.  DM = MF;   FN = NE
4. Congruent segments are segments of = length.
5.  DM + MF = DF;   FN + NE = FE
5. Segment Addition Postulate (or Whole Quantity)
6.  MF + DM = DF;   FN + FN = FE
6. Substitution
7.  2MF = DF;   2FN = FE
7. Addition (or Combine Like Terms)
8.  sasproof9;  sasproof10 
8. Multiplication (or Division) property of equality.
[This step establishes the ratio of similitude between the two triangles.]
9. SASproof4
9. Reflexive Property (or Identity Property)
10. SASproof5
10. SAS for Similar Triangles: If an ∠ of one Δ is congruent to the corresponding ∠ of another Δ and the lengths of the sides including these ∠s are in proportion, the Δs are similar.
11. sasproof6
11. Corresponding angles in similar triangles are congruent.
12. sasproof7
12. If 2 lines are cut by a transversal such that the corresponding angles are congruent, the lines are parallel.
13. sasproof8
13. Corresponding sides of similar triangles are in proportion. QED.



Proof of Mid-Segment Theorem - Using Parallelogram

For this proof, we will utilize an auxiliary line, congruent triangles and the properties of a parallelogram.
midPARA
midsegmentSproofGiven
Statements
Reasons
1. SASproof1b
1. Given
2. SASProof2a
2. A mid-segment joins the midpoints of two sides of a triangle.
3. Through E draw line parallel to df. Extend MN to intersect at M1.
3. Through a point not on a line, only one line can be drawn parallel to the given line. Parallel Postulate.
4. SASproof3
4. Midpoint of a segment divides a segment into 2 congruent segments.
5. DFE congruentFEM1
5. If 2 parallel lines are cut by a transversal, the alternate interior angles are congruent.
6.FNM congruent∠M1NE
6. Vertical angles are congruent.
7. ΔFNM congruentΔM1NE
7. ASA - If 2∠s and the included side of one Δ are congruent to the corresponding parts of another Δ, the Δs are congruent.
8. expara88
8. CPCTC - corresponding parts of congruent triangles are congruent.
9. expara88a
9. Substitution (or Transitive property)
10. DMM1E is a parallelogram
10. A quadrilateral with one pair of sides both || and congruent is a parallelogram.
11. expara99
11. A parallelogram is a quad. with 2 pair of opposite sides parallel.
12. expara99a
12. Opposite sides of a parallelogram are congruent.
13. expara99b
13. Congruent segments have = measure.
14. MN + M1N = MM1
14. Segment Addition Postulate (or whole quantity)
15. MN + MN = DE
15. Substitution
16. 2MN = DE
16. Addition (or combine like terms)
17. MN = ½DE
17. Division (or Multiplication) of Equalities



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