Construct a rhombus ABCD in which AB= 4 cm and angle ABC= 60 degree . Divide it into two triangle ABC and ADC. Construct a triangle AB 'C ' similar to triangle ABC with scale factor 2/3. Draw a line segment C 'D ' parallel to CD, where D ' lies on AD. Is AB 'C 'D ' a rhombus? Give reasons.

Following are the steps of construction:
(1) Construct a line segment AB of length 4 cm.

(2) Construct an angle of 60° at B.

(3) From B cut an arc of length 4 cm and name it as C. Join BC.

(4) From C and A, mark arcs of lengths 4 cm. Let they intersect at a point say D.

ABCD is the required rhombus.

(6) Now, construct an acute angle on the downward side of A. Name it as AX.

(7) Mark three points X1, X2 and X3 at equal distances on AX.

(8) Join X3B and construct a line X2B' parallel to X3B.

(9) Construct B'C' parallel to BC and C'D' parallel AB.

AB'C'D' is the required rhombus.

Justification:

In ΔABC,

By BPT

Similarly, in ΔACD,

By BPT

Thus, we have

As AB = CD

∴ AB'= CD'

Thus, AB'C'D' is a rhombus.

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