Question

abcd is a parallelogram and p and q are the mid-points of cd anf bc. if pq and ac intersect each other at R. prove that ac=4cr​

Answer 1

Answer:

Solution

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It is given that,

ABCD is a parallelogram

P and Q are mid-points of CD and BC

To prove: CR=

4

1

AC

Construction: Join AC and BD

Proof:

In parallelogram ABCD

Diagonals AC and BD bisect each other at O

⇒AO=OC

⇒OC=

2

1

AC ...(1)

In △BCD,

P and Q are mid points of CD and BC

⇒PQ∥BD[ By midpoint theorem ]

In △BCO

Q is the mid-point of BC and PQ∥OB

⇒R is the mid-point of CO

∴CR=

2

1

OC=

2

1

(

2

1

AC)

⇒CR=

4

1

AC

Hence, proved.

Answer 2

Step-by-step explanation:

It is given that,

ABCD is a parallelogram

P and Q are mid-points of CD and BC

To prove: CR=

4

1

AC

Construction: Join AC and BD

Proof:

In parallelogram ABCD

Diagonals AC and BD bisect each other at O

⇒AO=OC

⇒OC=

2

1

AC ...(1)

In △BCD,

P and Q are mid points of CD and BC

⇒PQ∥BD[ By midpoint theorem ]

In △BCO

Q is the mid-point of BC and PQ∥OB

⇒R is the mid-point of CO

∴CR=

2

1

OC=

2

1

(

2

1

AC)

⇒CR=

41

AC

Hence, proved.