Question

one of the diagonals of a rhombus is 12cm and area is 96sq cm the perimeter of rohmbus is

Answer 1

Answer:

• The perimeter of rhombus is 40 cm.

Solution:

Given:-

• One of the diagonals of a rhombus is 12 cm.
• Area of rhombus is 96 sq.cm.

To find:-

• The perimeter of rhombus.

Step-by-step explanation:

Formula:

• Area of rhombus = 1/2 × d₁ × d₂

Where,

• d₁ = 12 cm
• d₂ = x cm
• area = 96 sq.cm

Applying the values in the given formula,

$\\ \longmapsto\sf{96 = \frac{1}{\cancel{2}} \times \cancel{12} \times x}\\ \\ \longmapsto\sf{96 = 1 \times 6 \times x }\\ \\ \longmapsto\sf{96 = 6 \times x}\\ \\ \longmapsto\sf{x = \frac{\cancel{96}}{\cancel{6}}}\\ \\ \longmapsto\underline{\boxed{\mathfrak{x = 16}}}\; \green{\bigstar}$

• Therefore,The other diagonal(d₂) is 16 cm.

Now,finding the perimeter of rhombus,

• Perimeter of rhombus = 4 × side

To find the side of rhombs let us apply the pythagorean theorem.

• c = √a² + b²  

Where,

• a = d₁(12/2 = 6)
• b = d₂(16/2 = 8)
• c = x

Applying the values,

$\\ \longmapsto\sf{x=\sqrt{(6)^2 + (8)^2}} \\ \\ \longmapsto\sf{x=\sqrt{36 + 64}}\\ \\ \longmapsto\sf{x= \sqrt{100}}\\ \\ \longmapsto\boxed{\mathfrak{x=10}}\;\bigstar$

• The side of rhombus is 10 cm.

Then,

• Perimeter of rhombus = 4 × 10 = 40 cm.

Formula's used:-

• Area of rhombus = ½ × d₁ × d₂
• Perimeter of rhombus = 4a
• c = √a² + b²

Answer 2

$\frak{\maltese\:Given} = \begin{cases} &\sf{One \:of\: the\: diagonals \:of\: a \:rhombus \:is\: 12\: cm\:.} \\\\\\\\\\\\ &\sf{Area \:of\: rhombus\: is\: 96\: cm^{2}.}\end{cases}$

$\boxed{\underline{\underline{\texttt{\maltese\:\:To\:find:-\:\:}}}}$

$\qquad\sf{:\implies\:The\:perimeter\:of\:the\:rhombus\:.}$

$\boxed{\underline{\underline{\texttt{\maltese\:\:Assume:-\:\:}}}}$

$\qquad\sf{:\implies\:Let\:the\:second\:diognol\:of\:the\:rhombus\:be\:=\:x}$

$\boxed{\underline{\underline{\texttt{\maltese\:\:Solution:-\:\:}}}}$

$\maltese$ Finding the other diagonal of the rhombus .

$\color{fuchsia}{\qquad\sf{:\implies\:Area\:of\:_{(Rhombus)}\:=\:\dfrac{1}{2}\:\times\:d_{1}\:\times\:d_{2}}}$

$\qquad\sf{:\implies\:96\:=\:\dfrac{1}{2}\:\times\:12\:\times\:x}$

$\qquad\sf{:\implies\:96\:=\:1\:\times\:6\:\times\:x}$

$\qquad\sf{:\implies\:\dfrac{96}{1\:\times\:6}\:=\:x}$

$\qquad\sf{:\implies\:\dfrac{96}{6}\:=\:x}$

$\qquad\sf{:\implies\:x\:=\:16}$

$\color{lime}{\underline{\sf{Therefore\:the\:other\:digonal\:of\:Rhombus\:(d_{2})\:is\:16cm\:.}}}$

$\qquad\qquad\underline{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:}$

$\maltese$ Finding the perimeter of the rhombus .

Before finding the perimeter of the rhombus we have to find the side of the rhombus, So we will apply the Pythagorean theorem.

$\qquad\sf{:\implies\:c\:=\:\sqrt{a^{2}\:+\:b^{2}} }$

Here,

• a = d₁(12/2 = 6)
• b = d₂(16/2 = 8)
• c = x

$\qquad\sf{:\implies\:x\:=\:\sqrt{(6)^{2}\:+\:(8)^{2}} }$

$\qquad\sf{:\implies\:x\:=\:\sqrt{36\:+\:64}}$

$\qquad\sf{:\implies\:x\:=\:\sqrt{100}}$

$\sf{:\implies\:x\:=\:10 }$

$\color{red}{\underline{\sf{Therefore\:The \:side \:of \:rhombus \:is \:10 \:cm\:.}}}$

$\qquad\qquad\underline{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:}$

$\maltese$ Now the side of the rhombus is 10 cm ,so we will apply the formula [ Petrimter of the rhombus = 4 × side ] to find the perimeter .

$\qquad\sf{:\implies\:Perimeter\:of\:_{(Rohmbus)}\:=\:4\:\times\:side}$

$\qquad\sf{:\implies\:Perimeter\:of\:_{(Rohmbus)}\:=\:4\:\times\:10}$

$\qquad\sf{:\implies\:Perimeter\:of\:_{(Rohmbus)}\:=\:100\:cm\:.}$

$\color{aqua} {\underline{\sf{Therefore\:the\:perimeter\:of\:the\:rhombus\:is\:10\:cm\:.}}}$