Question

The perimeter of a rhombus with one diagonal 24 cm long is the same as the perimeter of an equilateral triangle with side 20 cm. Find the length of the other diagonal (in cm).

Answer 1

Hint: The sides of a rhombus are equal but the sides do not have an angle of 90 degree between them like a square has. Moreover, the diagonals intersect at 90 degrees.

Complete step-by-step answer:

We are given that the perimeter of a rhombus with one diagonal 24 cm long is the same as the perimeter of an equilateral triangle with side 20 cm.
Firstly we will find out the length of sides of a rhombus.
Let the side of the rhombus be equal to $l$ .
Since, the sides of a rhombus are equal the perimeter of a rhombus will be - $4l$ .
Now, we are given that the side of an equilateral triangle is 20cm and the perimeter of the triangle is equal to the perimeter of the rhombus. Therefore,
$3 \times 20 = 4l$
$4l = 60$
$l = 15 = AB$ ……. (1)
Since, the diagonals bisect each other \[AC = 24\] (given), we can get A0. Thus,
\[AO = 12{\text{ cm}}\] ………. (2)
Now, we take the triangle$AOB$.
Since the diagonals bisect each other at 90 degree we apply Pythagoras theorem:
$A{O^2} + O{B^2} = A{B^2}$
From equation (1) and (2) we have,
${(12)^2} + O{B^2} = {(15)^2}$
Subtracting both sides by ${(12)^2}$, we get,
$O{B^2} = {(15)^2} - {(12)^2}$
$O{B^2} = 225 - 144$
$O{B^2} = 81$
Taking square root both the sides we have,
$OB = 9$ ……. (3)
Now the diagonal BD will be twice the OB. Therefore,
\[BD = 2OB\]
From equation (3) we have,
\[BD = 2(9) = 18\]

Therefore, the length of the diagonal will be 18 cm.

Note: One mistake that students generally make is considering the length of diagonals of a rhombus as the same which is never true. The length of diagonals of a rhombus is not the same. The length of the diagonals are equal in Square.