Question

the diagonal AC and BD of a parallelogram ABCD intersect at O . If P is the mid point of AD. prove that (i) POllAB (ii) PO 1/2 of CD​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

ABCD is parallelogram.

Diagonals AC and BD intersects each other at O.

P is the midpoint of AD

We know that,

In parallelogram, diagonal bisects each other.

Since, AC and BD intersects each other at O.

It means, O is the midpoint of AC and BD.

Now, In triangle ACD

P is the midpoint of AD [ Given ]

O is the mid point of AC [Proved above ]

We know,

Midpoint Theorem :- This theorem states that line segment joining the midpoints of two sides of a triangle is parallel to third side and equals to half of it.

So, using Midpoint Theorem, we have

OP || CD and OP = 1/2 CD

Since, ABCD is a parallelogram, so AB || CD

So, it implies, OP || AB

Hence,

$\rm\implies \: \boxed{\sf{ \: \:OP \parallel \: AB \: and \: OP = \frac{1}{2} \: CD \: \: }}$

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Additional Information

Parallelogram have opposite pair of sides equal and parallel.

Opposite angles of a parallelogram are equal.

Diagonals bisect each other.

Answer 2

It is given that,

ABCD is a parallelogram in which diagonals AC and BD intersects each other at O, P is the mid point of AD.

To proff PQ || AB , PO 1/2 of CD

Proff:-

We know that,

The diagonals of parallelogram bisects each other

$\implies \: BO\:=\:OD$

$\implies O\:is\:the\: midpoint\:of\:BD$

$In\: \triangle ABD$

P and O is the mid point of AB and BD

PO || AB [ Mid point theorem ]

PQ || AB

Hence part 1 proved

To prove PO is 1/2 of CD we have to use mid point theorem :-

OP || CD and OP = 1/2 of CD

ABCD is a parallelogram so AB || CD

$\implies OP \: || \: AB$

$\color{aqua} \boxed{ \therefore \: OP \: || \: AB\: and \:OP\:=\: \frac{1}{2} \: of \: CD}$

Hence part 2 proved

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$\large\color{lime}\boxed{\colorbox{black}{Used\:Theorems}}$

$\color{red}{ \underline{❑ \: Converse \: of \: mid \: point \: theorem}}$

The converse of the midpoint theorem states that ” if a line is drawn through the midpoint of one side of a triangle, and parallel to the other side, it bisects the third side”.

$\color{pink}{ \underline{❑ \: Mid\:Point\:Theorem}}$

The midpoint theorem states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”

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