Three rectangles are to be cut along the diagonals and the triangles so got rearranged to form a regular pentagon, as shown in the picture. If the sides of the pentagon are to be 30 centimetres, what should be the length and breadth of the rectangles?

For the figures,

Given, WX = XY = YZ = ZY = YW = 30 cm

So, It can be easily seen that,

SQ = PN = WX = 30 cm [∵ Side of pentagon given]

CD = AB =

CD = AB =

Let’s say in rectangle PQRS,

∠PQS = ∠RSQ

[∵ Alternate interior angles for PQ ∥ SR on rectangle PQRS]

Similarly,

∠MNP = ∠OPN

∠RQS = ∠PSQ

[∵ Alternate interior angles for SP ∥ RQ on rectangle PQRS]

∠ONP = ∠MPN

So ∠WYJ = ∠ZYJ = ∠WXI = ∠YXI

And,

∠WYJ =

∠WYJ = [∵ Interior angle of regular pentagon = 108° ]

∠WYJ = 54°

So,

WJ = WY × sin54°

WJ = 30 × 0.81

WJ = 24.3 cm (i)

So, WZ = 2× WJ = 2× 24.3

WZ = 48.6

And, YJ = WY× cos54°

YJ = 30 × 0.59

YJ = 17.7 cm (ii)

Also, YZ = 2 × KZ

⇒ 30 = 2× KZ

⇒ KZ = 15 cm (iii)

By applying Pythagoras Theorem to ΔWKZ, we have,

WK² = WZ² - KZ²

⇒ WK² = 48.6² - 15²

⇒ WK² = 2361.96 – 225

⇒ WK² = 2136.96

⇒ WK = 46.2 cm (iv)

Thus dimensions of rectangles are as follows:

In ABCD,

Length AD = BC = 46.2 cm [From (iv) WK = 46.2 cm]

Breadth AB = CD = 15 cm [From (iii) KZ = 15 cm]

In PQRS and MNOP,

Length PS = QR = MP = NO = 24.3 cm [From (i) WJ = 24,3 cm]

Breadth PQ = SR = MN = PO = 17.7 cm [From (ii) YJ = 17.7 cm]