Question

X and Y are respectively the mid-point of sides AB and BC of a parallelogram ABCD. DX and DY intersect AC at M and N respectively. If AC=4.5 cm, find MN. ​

Answer 1

Step-by-step explanation:

ABCD is a parallelogram. X and Y are the mid points of the sides AB and BC respectively. DX and DY intersect AC at M and N respectively.

In Δ ABC, X and Y are mid points of AB and BC respectively.

∴ XY = 1/2AC and XY || AC

In Δ AOB, 

X is the mid point of AB and XS || OM  (XY || AC)

∴ S is the mid point of OB  (Converse of mid point theorem 

⇒ OS = OB

We know that, diagonals of the parallelogram bisect each other.

∴ OD = OB

⇒ OD = OS + SB

⇒ OD = 2OS  (OS = SB)

⇒ OD/OS = 2

⇒ OS/OD = 1/2

⇒ OS/OD + 1 = 1/2 + 1

⇒ (OS + OD)/OD = 3/2

⇒ DS/OD = 3/2

⇒ OD/DS = 2/3

Δ DMO ~ Δ DNO    (AA similarity)

∴ MO/XS = OD/DS 

⇒ MO/XS = 2/3

⇒ MO = 2/3XS  ..............(2)

Similarly ON = 2/3SY  ...........(3)

Adding (2) and (3), we get.

MO + ON = 2/3(XS + SY)

∴ MN = 2/3XY

⇒ MN = 2/3 × 1/2AC   (Using equation 1)

⇒ MN = 2/3 × 4.5/2

⇒ MN = 9/6

⇒ MN = 1.5 cm

Hope it helps!