In the rhombus ABCD, show that $4{\left( {AB} \right)^2} = {\left( {AC} \right)^2} + {\left( {BD} \right)^2}$

Last updated date: 18th Mar 2023
•
Total views: 306.6k
•
Views today: 6.89k
Answer
306.6k+ views
Hint – Take any one of the four triangles and apply PGT in it after using the few properties of the Rhombus, that is diagonals are perpendicular bisectors of each other and all sides are equal.
ABCD rhombus is shown above.
As we know in rhombus diagonals bisect each other and are perpendicular to each other.
$
\Rightarrow OB = OD,{\text{ & }}OA = OC..................\left( 1 \right) \\
\angle AOB = \angle AOD = \angle BOC = \angle DOC = {90^0} \\
$
And also we know that all the sides of the rhombus are equal.
$ \Rightarrow AB = BC = CD = DA$………………. (2)
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}$…………. (3)
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 (3)
$
\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} \\
$
Hence Proved.
Note – In such types of questions the key concept we have to remember is that always recall the condition of Rhombus which is stated above in equation (1) and (2), then use the property of Pythagoras theorem which is also stated above and simplify according to properties of rhombus, we will get the required result.
ABCD rhombus is shown above.
As we know in rhombus diagonals bisect each other and are perpendicular to each other.
$
\Rightarrow OB = OD,{\text{ & }}OA = OC..................\left( 1 \right) \\
\angle AOB = \angle AOD = \angle BOC = \angle DOC = {90^0} \\
$
And also we know that all the sides of the rhombus are equal.
$ \Rightarrow AB = BC = CD = DA$………………. (2)
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}$…………. (3)
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 (3)
$
\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} \\
$
Hence Proved.
Note – In such types of questions the key concept we have to remember is that always recall the condition of Rhombus which is stated above in equation (1) and (2), then use the property of Pythagoras theorem which is also stated above and simplify according to properties of rhombus, we will get the required result.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
