Question

side of a rhombous is 10 cm and one of it's diagonal is 16 cm. find length of other diagonal.​

Answer 1

★$~\large \underline{\sf{\pink{Question:}}}$

Given that, the side of a rhombus is 10cm and one of its diagnol is 16cm.

★$~\large \underline{\sf{\pink{To~ find:}}}$

We have to find the length of other diagnol.

★$~\large \underline{\sf {\pink{Explaination:}}}$

We are given the length of one side of a rhombus and the length of one of its diagnol. To solve this, we should be aware of the basic properties of a rhombus like diagnols bisect each other perpendicularly and all the sides are equal in a rhombus. Using these properties, we'll be able to find the length of the second diagnol.

★$~\large \underline{\sf {\pink{Solution:}}}$

Let PQRS be the required rhombus with :

• $\sf PS=SR=RQ=QP=10cm$
• $\sf PR=16cm$
• $\sf PO=8cm$

Since the diagnols get bisected perpendicularly, we get a right angled $\sf \triangle POQ$ with $\sf \measuredangle O=90\degree.$

From Pythagoras theorem, we get :

$\sf :\implies ~PO^2~+~SO^2~=~PS^2$

$\sf :\implies ~ (8cm)^2~+ SO^2~ = ~(10cm)^2$

$\sf :\implies ~ 64cm^2~ +~SO^2~=~ 100cm^2$

$\sf :\implies~ SO^2~=~ 100cm^2~-~64cm^2$

$\sf :\implies ~ SO^2~=~ 36cm^2$

$\sf :\implies ~ SO~=~\sqrt{36cm}$

$\pink{\sf{ :\implies ~SO~=~6cm }}$

We know that diagnols in a rhombus bisect each other and from this we get :

$\sf :\implies ~Diagnol~QS~=~SO~+~OQ$

$\sf :\implies~ Diagnol~QS~=~SO~+~SO$

$\sf :\implies ~Diagnol~QS~=~2(SO)$

$\sf :\implies~ Diagnol~QS~=~2(6cm)$

$\sf\pink{ :\implies ~Diagnol~ QS~=~12cm}$

______________________

$\bf {\therefore~ The~ length~ of~ other~ diagnol~ is~12cm.}$

Answer 2

Given: Each side of the rhombus is 10cm & one of its diagonal is 16cm.

Need to find: Length of the other diagonal.

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\underline{\bf{\dag\;}\frak{Using\;Pythagoras\;theorem:}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\star\;\boxed{\pink{\sf{(Base)^2+(Perpendicular)^2=(Hypotenuse)^2}}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

where,

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• Hypotenuse = AB = 10cm
• Base = BO
• Perpendicular = AO = 8cm

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

Therefore,

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$:\implies\sf{(BO)^2+(AO)^2=(AB)^2}\\\\\\\\:\implies\sf{(BO)^2+(8)^2=(10)^2}\\\\\\\\:\implies\sf{(BO)^2=100-64}\\\\\\\\:\implies\sf{(BO)^2=36}\\\\\\\\:\implies\sf{BO=\sqrt{36}}\\\\\\\\:\implies\underline{\boxed{\purple{\frak{BO=6\;cm}}}}{\;\bigstar}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

Hence,

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

• Length of the other diagonal = BD = 6 + 6 = 12cm

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$\therefore\;{\underline{\sf{Hence,\;the\;length\;of\;the\;other\;diagonal\;is\;{\textsf{\textbf{12\;cm}}}}.}}$⠀⠀