The perimeter of a rhombus is 40 cm. If the length of one of its diagonals is 12 cm, what is the length of the other diagonal?

Grade : 9

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16 cm

8 cm

20 cm

10 cm

answer is A.

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Detailed Solution

Let us consider the length of aside of a rhombus as 'a'.
We know that all the sides of a rhombus are equal to each other.
So, the length of each side is 'a'.
The perimeter of a rhombus with side 'a' is given by the formula(a+a+a+a ).
So, the value of the perimeter of the rhombus is equal to '4a'.
It is given that ,
the perimeter of the rhombus=40cm.
So , a = 40 ÷ 4 = 10 cm .
Question Image

We have a rhombus ABCD in which AD = 10cm
AC = 12 cm.
We know that ,
in a rhombus the diagonals bisect each other at 90∘.
Now, use Pythagoras theorem in
Δ AOD
and get the value of OD.
BD gets bisected at the point O.
So, BD=2×OD.
Now,
get the value of the diagonal BD.
The length of the diagonal AC
= 12 cm.
Since the diagonal AC gets bisected at the point O
so, the length OA will be half of the diagonal AC.
OA=AC/2
=12/2
=6cm .
Now, in Δ AOD
we have,
OA= 6 cm
AD = 10 cm
Now, using Pythagoras theorem,
⇒(Hypotenuse)²=(Base)²+(Height)²⇒(AD)²=(OA)²+(OD)²
⇒(10)²=(6)²+(OD)²
⇒100=36+(OD)²
⇒100−36=(OD)²
⇒64=(OD)²
⇒√64= OD
⇒OD=8
The diagonals AC and BD are bisecting at point O.
So, BD=2×OD=2×8=16
.
Therefore, the length of the diagonal BD is 16 cm.
So, option (A) is the correct answer.

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