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A rhombus of side 20cm each. Find the length of each diagonal.

A rhombus of side 20cm each. Find the length of each diagonal.

Grade:10

7 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
9 years ago
Hello student,
Please find the answer to your question below
IfABCDis a rhombus with side lengthxthen, by the parallelogram law, and because we know each side has equal length, we know that4x2=|AC|2+|BD|2 where|AC|is the length of the diagonal fromAtoCand similarly for|BD|.
As every side is equal it becomes a square
SoIn the case of a square, we get|AC|=|BD|and so2x2=|AC|2⇒|AC|=2√⋅x.
So length of diagonal AC=BD=2\sqrt{20}=4\sqrt{5}cm
Siddharth Gurla
16 Points
9 years ago
Thank you Sir for your response. However I have a few doubts related to this solution.
 
  • What is the parallelogram law ?
  • How does it become a square. As the figure given is a rhombus.
  • Lastly , I think that for a square there is more criteria than the equality of sides.
Thanks.
SHAIK AASIF AHAMED
askIITians Faculty 74 Points
9 years ago
Hello student,
Actually there was some misinterpretation in the above answer.
Please find your answer below
IfABCDis a rhombus with side length 20cm then if we draw diagonals.As the diagonals bisect each other at right angles they form 4 pythagorean triangles.
So if we apply pythagoras theorem to each triangle we get

240-1185_rhombus_39.GIF
in triangles AOD,COD,BOC,AOB we get the lengths of OA,OB,OC,OD as 10\sqrt{2}
So AC=OA+OC=20\sqrt{2}cm
BD=OB+OD=20\sqrt{2}cm
Siddharth Gurla
16 Points
9 years ago
Sir,
We seemed to have based the solution on the asssumption that the diagonals of a rhombus are equal. I am aware of the diagonals bisecting at right angles only. Kindly clarify , if I am missing something.
 
Thanks. 
SHAIK AASIF AHAMED
askIITians Faculty 74 Points
9 years ago
Hello student,
Please find my response to your question below
It is not possible to find out the length of diagonals if they are not assumed to be same.
If the lengths of diagonals are different then atleast the length of one diagonal should be given.
Siddharth Gurla
16 Points
9 years ago
Thanks very much for your reply Sir. 
Sayan
35 Points
9 years ago
The diagonals of a rhombus are equal iff the rhombus is a square. Hence the answer cant be found out if we do not assume it to be equal 

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