Midsegment of a Triangle

What is Midsegment of a Triangle

The midsegment of a triangle is a line connecting the midpoints or center of any two (adjacent or opposite) sides of a triangle. It is parallel to the third side and is half the length of the third side.

How Many Midsegments Does a Triangle Have

Since a triangle has three sides, each triangle has 3 midsegments. In the given ∆ABC, DE, EF, and DF are the 3 midsegments. The 3 midsegments form a smaller triangle that is similar to the main triangle. Thus, ∆ABC ~ ∆FED

Properties

1. It joins the midpoints of 2 sides of a triangle; in ∆ABC, D is the midpoint of AB, E is the midpoint of AC, & F is the midpoint of BC
2. A triangle has 3 possible midsegments; DE, EF, and DF are the three midsegments
3. The midsegment is always parallel to the third side of the triangle; so, DE ∥ BC, EF ∥ AB, and DF ∥ AC
4. The midsegment is always 1/2 the length of the third side; so, DE =1/2 BC, EF =1/2 AB, and DF =1/2  AC

Formula

How to Find the Midsegment of a Triangle

The formula to find the length of midsegment of a triangle is given below:

Triangle Midsegment Theorem

Proof of Triangle Midsegment Theorem

To prove: DE ∥ BC; DE = ½ BC

Proof: A line is drawn parallel to AB, such that when the midsegment DE is produced it meets the parallel line at F

Given: D is the midpoint of AB

E is the midpoint of AC

F is the midpoint of BC

Steps	Statement	Reason
1.	In ∆ADE and ∆CFE AE = EC ∠AED  = ∠CEF ∠DAE  = ∠ECF	E is the midpoint of AC Vertically opposite angle Alternate angles
2.	∆ADE ≅ ∆CFE	By AAS congruency of triangle
3.	DE = FE AD = CF	Corresponding parts of Congruent triangles (CPCTC)  are congruent
4.	AD = BD BD = CF	D is the midpoint of AB
5.	DF ∥ BC and DF = BC DE ∥ BC and DF = BC DE = ½ DF	DBCF is a paralleogram
6.	DE =1/2 BC	DF = BC Hence Proved

Converse of Triangle Midsegment Theorem

Converse of Triangle Midsegment Theorem Proof

To prove: DE is the midsegment of ∆ABC

AD = DB; AE = EC

Proof:

Given: D is the midpoint of AB

E is the midpoint of AC

DE ∥ BC

DE = 1/2 BC

Steps	Statement	Reason
1.	AD = DB AB = AD + DB = DB + DB = 2DB	D is the midpoint of  AB
2.	DE ∥ BC BD ∥ CF	DBCF is a parallelogram
3.	BD = CF DA = CF	Opposite sides of a parallelogram are equal
4.	In ∆ADE and ∆CFE, DA = CF	∠AED  = ∠CEF (Vertically opposite angle)∠DAE  = ∠ECF (Alternate angles)
5.	∆ADE ≅ ∆CFE	By AAS congruency of triangle
6.	AE = EC (E is the midpoint of AC) Similarly, AD = DB (D is the midpoint of AB) DE is the midsegment of ∆ABC	Corresponding parts of Congruent triangles (CPCTC)  are congruent

Solved Examples