Question

A rhombus ha a perimeter of 80 meters and the length of one diagonal is 24 meters.find the area of the rhombus

Answer 1

Answer:

The area of the rhombus is $A=384\ m^2$

Step-by-step explanation:

we know that

A rhombus have all four sides congruent.

The diagonals bisect each other

The diagonals are perpendicular

see the attached figure to better understand the problem

step 1

Find the length side of the rhombus

Divide the perimeter by 4

Let

b ---> the length side of rhombus

$b=\frac{P}{4}$

$b=\frac{80}{4}=20\ m$

step 2

Find the length of the other diagonal BD

In the right triangle AOB

Let

AC =24\ m ----> the length of one diagonal

BD ---> the length of the other diagonal

we have

$AO=24/2=12\ m\\AB=20\ m$

$AB^2=AO^2+OB^2$

substitute the values

$20^2=12^2+OB^2$

$OB^2=400-144\\OB=16\ m$

$BD=2(OB)=2(16)=32\ m$

step 3

Find the area of the rhombus

The area of the rhombus is equal to

$A=\frac{1}{2}(AC)(BD)$

substitute

$A=\frac{1}{2}(24)(32)=384\ m^2$

Answer 2

GIVEN:-

• $\rm{Area\:of\:Rhombus = 80m}$
• $\rm{Diagonal\:of\:Rhombus = 24m}$

TO FIND :-

• The area of Rhombus.

CONSTRUCTION:-

• Draw another Rhombus from B to D.

CONCEPT USED

• All Sides of Rhombus are equal.

• Diagonal Bisect each other at right angled.

FORMULAE USED:-

• ${\boxed{\rm{\blue{Area\:of\:Rhombus = \dfrac{Product\:of\:Diagonal}{2}}}}}$

• ${\boxed{\rm{\red{Pythogoras\:Theorem}}}}$

Now,

$\rm \implies{Perimeter\:of\:Rhombus = 4\times{sides}}$

$\rm \implies{80 = 4\times{sides}}⟹80=4×sides$

$\implies\rm{Side = \dfrac{80}{4}}$

$\implies\rm{Side = 20m}$

Now, In right angled triangle ∆OBC,

$\implies\rm{OC = 12m}$

$\implies\rm{BC = 20m}$

$\implies\rm{OB = ?}$

Using Pythogoras Theorem,

$\implies\rm{(OB)^2+(OC)^2 = (AB)^2}$

$\implies\rm{(OB)^2 = (20)^2-(12)^2}$

$\implies\rm{(OB)^2 = \sqrt{256}}$

$\implies\rm{OB = 16m}$

Therefore,

$\implies\rm{2OB = BD}$

$\implies\rm{2\times{16} = BD}$

$\implies\rm{32 = BD}$

Now,

$\rm{Area\:of\:Rhombus = \dfrac{Product\:of\:Diagonal}{2}}$

$\rm{Area\:of\:Rhombus = \dfrac{32\times{24}}{2}}$

$\rm{Area\:of\:Rhombus = \dfrac{768}{2}}$

$\rm{Area\:of\:Rhombus = 384m^2}$

$\sf \therefore Hence, \: The \: Area \: of \: Rhombus \: is \: 384m².$