Question

State and prove midpoint therom

Answer 1

A midpoint is a point on a line segment equally distant from the two endpoints. The Midpoint Theorem is used to make a bold statement regarding triangle sides and their lengths. Given a triangle, if we connect two sides with a line segment, and this line segment joins each of the two sides at the centers, or midpoints of each side, we can know two very important aspects about the triangle and the relationships between the sides

Anytime you have a line segment that connects two sides of a triangle at the midpoints, you automatically know that the sides are cut in half, and that the segment is parallel to the third side of the triangle. Parallel sides are shown by using this symbol ||. You also know the line segment is one-half the length of the third side.

Answer 2

Statement :-

The midpoint theorem states that when the midpoints of any two sides of a triangle is joined, it is parallel and half of the third side.

Note:-

Refer the diagrams in the attachments to understand all the steps.

We are considering right angled triangle in this case. You can use this in all sorts of triangles.

Proof :-

Let us construct ray parallel to AB from C and extend PQ to Q'.

As per observation, PB║CQ'.

AQ = QC

∠AQP = ∠CQQ' (Vertically opposite angle)

∠PAQ = ∠QCQ' (Alternately equal angle)

∴ ΔAQP ≅ ΔCQQ' (AAS criteria)

    AP = Q'C (CPCT)

    PQ = QQ' (CPCT)

Now,

$\frac{AP}{AB} = \frac{AQ}{AC} = \frac{1}{2}$

∠BAC = ∠PAQ  (common)

∴ ΔBAC = ΔPAQ

Hence, $\frac{AP}{AB} = \frac{AQ}{AC} =\frac{PQ}{BC} = \frac{1}{2}$

and 2QQ'=PQ'

PQ'CB is a rectangle.

So,

PQ' ║ BC or PQ║BC.

2PQ = BC.

QED