Question

What is mid point theorem?

Answer 1

Midpoint theorem states that if you join midpoints of any two sides, it will be parallel to the third side and half of the third side.
Let us take a triangle ABC. Let D be the mid-point of AB and E be the mid-point of BC.
Then, DE will be parallel to BC and DE = $\frac{1}{2}$ BC.

Contruction: Produce DE to F such that DE=EF
Proof:
In $\Delta ADE$ and $\Delta CFE$
AE = EC ...(given, E is the midpoint)
DE = EF ...(by construction)
$\angle AED=\angle CEF$ ...(vertically opposite $\angle$s)
$\therefore \Delta ADE\cong \Delta CFE$
$\therefore AD=CF$ ...(i)
and $\angle ADE=\angle CFE$ ....(CPCT)
Since AD = DB ...(D is the mid-point)
and AD = CF ... (proved above)
$\therefore$ DB = CF ...(ii)
Since $\angle ADE=\angle CFE$
$\therefore$ AD || CF
$\therefore$ AB || CF
$\therefore$ DB || CF ...(iii)
Now in quadrilateral BCFD,
DB = CF and DB || CF
$\therefore$ BCFD is a parallelogram
BC = DF = DE + EF = 2 DE
$\therefore$ DE = BC
Also, BC || DF ...(opposite sides of ||gm)
$\therefore$ BC || DE and DE || BC

So, BE || BC and DE $=\frac{1}{2}BC$