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The length of the diagonals of a rhombus are $6cm$ and $8cm$ . Find the length of each side of the rhombus.
(a) $2cm$
(b) $3cm$
(c) $4cm$
(d) $5cm$

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Last updated date: 13th Jun 2024
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Answer
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Hint: In the above question, we are given a rhombus with the length of the both diagonals. Now, we know that rhombus is a quadrilateral which has all the sides equal and the opposite angles are equal. Now, we know that the radius is half of diameter. We will use this to find the lengths of some sides. Later we will use the pythagoras theorem to find the length of the sides.

Complete step-by-step solution:
Now, the given figure denotes the rhombus
seo images

So,\[AC=8cm\] and \[BD=6cm\]
Now, we know that the diagonals of rhombus divide each other in two equal halves.
Hence, \[AO=4cm\] and \[OD=6cm\]
Now, we know that diagonals of rhombus bisect each other perpendicularly.
So, we can say that \[\angle AOD=90{}^\circ \]
In right \[\Delta AOD\] ,
We know that, \[AO=4cm\] and \[OD=3cm\]
So, by using the Pythagoras Theorem,
$\Rightarrow A{{D}^{2}}=A{{O}^{2}}+O{{D}^{2}}$
$\Rightarrow A{{D}^{2}}={{\left( 4 \right)}^{2}}+{{\left( 3 \right)}^{2}}$
$\Rightarrow AD=5cm$
So, \[AD\] is \[5cm\]
So, the sides of the rhombus are of length \[5cm\]

Hence, the correct option is \[d\].

Note: Remember the properties of Rhombus:
All sides of the rhombus are equal.
The opposite sides of rhombus are parallel.
Opposite angles of rhombus are equal.
In a rhombus, diagonals bisect each other at right angles.
Diagonals bisect the angles of a rhombus.
The sum of two adjacent angles is equal to \[180\] degrees.
Diagonals of rhombus divide each other in two equal halves.