Question

The lengths of the diagonal of a rhombus are 24 cm and 10 cm. Find each side of the rhombus.

Answer 1

Hint: We need to find each side of rhombus, use the property of rhombus that its diagonals bisect each other. This will help in formation of 4 different right angle triangles whose hypotenuse can be evaluated, this will help to get the answer.

Complete step-by-step answer:

Let ABCD be a quadrilateral as shown above.

And we know that the diagonals of the rhombus bisect each other i.e. it is perpendicular to each other, (i.e. it cuts the diagonals into two equal parts).
$
   \Rightarrow OB = OD,{\text{ & }}OA = OC..................\left( 1 \right) \\
  \angle AOB = \angle AOD = \angle BOC = \angle DOC = {90^0} \\
$

So in triangle AOB apply Pythagoras Theorem
${\left( {{\text{Hypotenuse}}} \right)^2} = {\left( {{\text{Perpendicular}}} \right)^2} + {\left( {{\text{Base}}} \right)^2}$
$ \Rightarrow {\left( {AB} \right)^2} = {\left( {OA} \right)^2} + {\left( {OB} \right)^2}$…………. (2)

Now from figure $AC = OA + OC,{\text{ }}BD = BO + OD$
From equation (2)
$
  AC = OA + OA,{\text{ }}BD = OB + OB \\
   \Rightarrow OA = \dfrac{{AC}}{2},{\text{ }}OB = \dfrac{{BD}}{2} \\
 $

Now from equation (2)
$
   \Rightarrow {\left( {AB} \right)^2} = {\left( {\dfrac{{AC}}{2}} \right)^2} + {\left( {\dfrac{{BD}}{2}} \right)^2} \\
   \Rightarrow {\left( {AB} \right)^2} = \dfrac{{{{\left( {AC} \right)}^2}}}{4} + \dfrac{{{{\left( {BD} \right)}^2}}}{4} \\
 $
$ \Rightarrow 4{\left( {AB} \right)^2} = {\left( {AC} \right)^2} + {\left( {BD} \right)^2}$…………………….. (3)
Similarly,
$ \Rightarrow 4{\left( {BC} \right)^2} = {\left( {AC} \right)^2} + {\left( {BD} \right)^2}$……………………… (4)
$ \Rightarrow 4{\left( {CD} \right)^2} = {\left( {AC} \right)^2} + {\left( {BD} \right)^2}$………………………… (5)
$ \Rightarrow 4{\left( {DA} \right)^2} = {\left( {AC} \right)^2} + {\left( {BD} \right)^2}$…………………………… (6)

Now from equation (3), (4), (5) and (6)

AB = BC = CD = DA

Therefore all the sides of the rhombus are equal.

Now it is given that AC = 24 cm and BD = 10 cm.

So from equation (1) we have,
$ \Rightarrow 4{\left( {AB} \right)^2} = {\left( {24} \right)^2} + {\left( {10} \right)^2} = 576 + 100 = 676$

Now divide by 4 throughout we have,
$ \Rightarrow {\left( {AB} \right)^2} = 169 = {\left( {13} \right)^2}$
$ \Rightarrow AB = 13$ cm

Therefore the length of each side of the Rhombus is 13 cm.

So, this is the required answer.

Note: Whenever we face such types of problems the key concept is to have a diagrammatic representation of the figure as it helps understanding the geometry of the figure. The property of the rhombus helps getting on the right track to get the answer, so it is advisable to have a good gist of these properties.