Question

Three rectangles are cut along their diagonals and the triangles so got are rearranged to form a regular pentagon as shown below:

Find the dimensions of the rectangles.

Answer 1

Given: ABCDE is a regular pentagon made by using some pieces of rectangles cut along its diagonal.

We know that all the sides of a regular pentagon are equal.

∴ AB = BC = CD = DE = AE = 30 cm

Measure of each angle of a pentagon = 108°

∴ ∠AED = 108°

Measure of each angle of a rectangle = 90°

⇒ ∠AHE = ∠DHE = 90°

In ΔAED:

AE = DE

∴ ∠EDA = ∠EAD (Angles opposite to equal sides are equal in measure.)

Using angle sum property in ΔAED:

∠AED + ∠EDA + ∠DAE = 180°

⇒ 108° + 2∠EDA = 180°

⇒ 2∠EDA = 180° − 108°

⇒ 2∠EDA = 72°

⇒ ∠EDA = 36°

In ΔEHD:

Therefore, the dimensions of the smaller rectangle are 24.27 cm and 17.63 cm.

Now, ∠ADC = ∠EDC − ∠EDH

= 108° − 36°

= 72°

CF = FD = =15 cm

In ΔAFD:

Therefore, the dimensions of the larger rectangle are 46.165 cm and 15 cm.