Question

if the sides of a rhombus are 5 cm each and one diagonal is 8 cm, calculate
(1) the length of the other diagonal, and (ii) the area of the rhombus.​

Answer 1

Step-by-step explanation:

Diagram:- Refer the attachment

Given:-

• Each side of a rhombus = 5cm

• One of the diagonals of the rhombus = 8cm.

To Find:-

• The length of the other diagonal.

• The area of the rhombus.

Solution:-

(i) As we know that:-

For a rhombus :-

$\sf \implies {Side}^{2} = \bigg(\dfrac{ d_{1}}{2} \bigg)^{2} + \bigg(\dfrac{ d_{2}}{2} \bigg)^{2}$

Where, d₁ and d₂ are the diagonals.

$\sf \implies {(5)}^{2} = \bigg(\dfrac{ 8}{2} \bigg)^{2} + \bigg(\dfrac{ d_{2}}{2} \bigg)^{2}$

$\sf \implies {(5)}^{2} = {(4)}^{2} + \dfrac{ (d_{2} )^{2} }{ {(2)}^{2} }$

$\sf \implies 25 = 16 + \dfrac{ (d_{2} )^{2} }{ {4} }$

$\sf \implies \dfrac{ (d_{2} )^{2} }{ {4} } = 25 - 16$

$\sf \implies (d_{2} )^{2} = 9 \times 4$

$\sf \implies (d_{2} )^{2} =36$

$\sf \implies d_{2} = \sqrt{36} = 6$

$\sf \therefore \underline{\:\: \underline{\: Length \: of \: other \: diagonal \: (d_{2})= 6cm\:}\:\:}$

$\sf (ii) \: Area \: of \: rhombus = \dfrac{1}{2}\times d_{1} \times d_{2}$

$\sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \dfrac{1}{2}\times 8 \times 6$

$\sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \dfrac{48}{2}$

$\sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 24cm^2$

$\large \underline{\boxed{\sf \therefore Area \: of \: the \: rhombus = 24cm^2}}$

Answer 2

Answer:

1.Diagonals bisect each other and they are perpendicular in rhombus

Let half the length of second diagonal be x

As two diagonals and a side form right angled triangle ;

= ײ + 4²=5²

= x = 3

2.⇒length of 2nd side =2x=6cm

Area of rhombus =  1/2d₁d₂

                          = 1/2*8*6

​                     = 24 cm²

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