A wire is in the shape of a regular pentagon of side 12 cm. It is rebent into the shape of a rectangle whose length is $\dfrac{3}{2}$ times its breadth. Find the length and the breadth of the rectangle.

Given:

Side of regular pentagon = 12 cm

Length of rectangle = $\dfrac{3}{2}$ × breadth

Length of the wire = Perimeter of regular pentagon

= 5 × side

= 5 × 12 cm

= 60 cm

Let the breadth of the rectangle be b cm.

Then, length of the rectangle = $\dfrac{3}{2}$ b cm

Since the wire is rebent into the shape of a rectangle,

Perimeter of rectangle = Length of wire

⇒ 2(length + breadth) = 60

⇒ 2 $\Big(\dfrac{3}{2}b + b\Big)$ = 60

⇒ 2 × $\dfrac{3b + 2b}{2}$ = 60

⇒ 5b = 60

⇒ b = $\dfrac{60}{5}$

⇒ b = 12 cm

Length of rectangle = $\dfrac{3}{2}$ × 12 = 18 cm

Hence, the length of the rectangle is 18 cm and the breadth is 12 cm.