what is difference between mid point and converse mid point theorem?
Answers
Geometry is an essential and fundamental branch of mathematics. It deals with geometrical figures and problems based on their properties. There is one of the basic theorems in geometry which explains about an important property of triangle. This theorem is known as midpoint theorem.
The midpoint theorem states that:
"In a triangle, the line segment that joins the midpoints of the two sides of a triangle is parallel to the third side and is half of it."Explanation:
If the midpoints of any sides of a triangle are joined by a line segment, then this line segment will be parallel to the remaining side and will measure half of the remaining side.
Let us consider ABC is a triangle as shown in the following figure:
Let D and E be the midpoints of AB and AC. Then line DE is parallel to BC and DE is half of BC; i.e.
DE ∥∥ BC
DE = 1212 BC.
Midpoint theorem plays a vital role in mathematics. Let us learn more about midpoint theorem, its converse and its applications in detail.
The proof of mid point theorem is as follows.
Have a look at the following diagram:
Here, In △△ ABC, D and E are the midpoints of sides AB and AC respectively. D and E are joined.
Given: AD = DB and AE = EC.
To Prove: DE ∥∥ BC and DE = 1212 BC.
Construction: Extend line segment DE to F such that DE = EF.
Proof: In △△ ADE and △△ CFE
AE = EC (given)
∠∠AED = ∠∠CEF (vertically opposite angles)
DE = EF (construction)
hence
△△ ADE ≅≅ △△ CFE (by SAS)
Therefore,
∠∠ADE = ∠∠CFE (by c.p.c.t.)
∠∠DAE = ∠∠FCE (by c.p.c.t.)
and AD = CF (by c.p.c.t.)
The angles ∠∠ADE and ∠∠CFE are alternate interior angles assuming AB and CF are two lines intersected by transversal DF.
Similarly, ∠∠DAE and ∠∠FCE are alternate interior angles assuming AB and CF are two lines intersected by transversal AC.
Therefore, AB ∥∥ CF
So, BD ∥∥ CF
and BD = CF (since AD = BD and it is proved above that AD = CF)
Thus, BDFC is a parallelogram.
By the properties of parallelogram, we have
DF ∥∥ BC
and DF = BC
DE ∥∥ BC
and DE = 1212BC (DE = EF by construction)
Hence proved.
ConverseBack to Top
The converse of midpoint theorem does also exist. In geometry, the converse of midpoint theorem is as important as the theorem itself.
The converse of midpoint theorem states that:
"If a line segment is drawn passing through the midpoint of any one side of a triangle and parallel to another side, then this line segment bisects the remaining third side.
Explanation:
Have a look at the figure given below:
Consider the following triangle △△ABC, in which D is the midpoint of AB. A line segment DE is drawn which meets AC in E and is
parallel to the opposite side BC. In this case, the third AC is bisected by the line segment DE; i.e. we can say that
if AD = DB and DE ∥∥ BC
then AE = EC.
ExamplesBack to Top
Few example problems of midpoint theorem are given below.
Example 1: Find the value of x from the following diagram:
Solution: Here, given that AP = PB and BQ = QC
Thus, by midpoint theorem -
PQ ∥∥ AC and
PQ = 1212 AC
6 = 1212 x
x = 12
Example 2: Determine the value of x from the following image:
Solution: Here, given that AP = PC and BQ = QC
Thus, by midpoint theorem:
PQ ∥∥ AB and
PQ = 1212 AB
x = 1212 6
x = 3
Example 3: Find the value of x.
If the midpoints of any sides of a triangle are joined by a line segment, then this line segment will be parallel to the remaining side and will measure half of the remaining side.
Converse :
If a line segment is drawn passing through the midpoint of any one side of a triangle and parallel to another side, then this line segment bisects the remaining third side.