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The diagonals of a rhombus are 10cm and 24cm. Find the length of each side of a rhombus.

Answer
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Hint: Let us draw a rough figure as follows
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We solve this problem by using the definition of rhombus that is all the sides of rhombus are equal. We also use the condition that the diagonals intersect at right angles at mid – points. For finding the side length we use the Pythagoras Theorem states that the square of hypotenuse is equal to sum of squares of other two sides that is for the triangle shown below
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The Pythagoras theorem is given asb2=a2+c2.

Complete step by step answer:
We are given that the length of diagonals as 10cm and 24cm
Let us assume that from the figure the length of diagonals as
AC=24cm
BD=10cm
We know that the diagonals of rhombus intersect at right angles at mid – points that means the point ‘E’ is mid – point of both ‘AC’ and ‘BD’
By using the above condition the length of ED can be calculated as
ED=BD2
By substituting the required values we get
ED=10cm2=5cm
 Similarly, by using the condition that point ‘E’ is mid – point of ‘AC’ we get
AE=AC2
By substituting the required values we get
AE=24cm2=12cm
Now, let us consider the triangle ΔAED
We know that the Pythagoras Theorem states that the square of hypotenuse is equal to sum of squares of other two sides that is for the triangle shown below
seo images

The Pythagoras theorem is given asb2=a2+c2.
By using the Pythagoras theorem to triangle ΔAED we get
AD2=AE2+ED2
By substituting the required values in above equation we get
AD2=52+122AD2=25+144AD=169=13cm
We know that all the sides of a rhombus are equal
By using the above condition we get
AB=AC=CD=DA=13cm

Therefore the length of each side of a given rhombus is 13cm.

Note: We can solve this problem in another method.
If p,q are lengths of diagonals of a rhombus then the area is given as
A=12×p×q
If p is length of diagonal and a is side length of rhombus the area formula is given as
A=12×p×4a2p2
Since the area will be same in any method by equating them we get
12×p×q=12×p×4a2p2q=4a2p2
Now by squaring on both sides we get
4a2=p2+q2
Now, by substituting the required values we get
4a2=102+122a2=100+1444a=169=13cm
Therefore the length of each side of a given rhombus is 13cm.
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