Question

what is mid point theroem prove it

Answer 1

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.

ProofBack to Top
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.

Answer 2

Answer:

Converse of mid point theorem

Refer to the attachment_

Step-by-step explanation:

$hope \: it \: helps$