Question

Thhe diagonals of a rhombus are in ratio 3:4 if the longer diagonal is 12cm then find the are of rhombus

Answer 1

S O L U T I O N

Diagonals of a rhombus are in the ratio of 3:4. And, the longer diagonal is 12 cm.

We've to find out the area of the rhombus.

let's consider that smaller & longer diagonal of the rhombus be 3x & 4x.

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$:\implies\sf 4x = 12 \\\\\\:\implies\sf x = \cancel\dfrac{12}{4} \\\\\\:\implies\sf\pink{ x = 3}\\\\\\:\implies\sf 3x \qquad\qquad \bigg\lgroup\bf Smaller \ Diagonal \bigg\rgroup \\\\\\:\implies\sf 3 \times 3\\\\\\:\implies\boxed{\bf{\blue{Diagonal_{(smaller)} = 9 \: cm}}}\\\\\\:\implies\sf 4x \qquad\qquad \bigg\lgroup\bf Longer \ Diagonal \bigg\rgroup\\\\\\:\implies\sf 4 \times 3\\\\\\:\implies\boxed{\bf{\blue{Diagonal_{(longer)} = 12 \: cm}}}$

⠀By using the formula,

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$\star\ \boxed{\purple{\sf{Area_{(rhombus)} = \frac{1}{2} \times (d_1) \times (d_2)}}}$

$\bf{Diagonals}\begin{cases}\sf{d_{1} = 9 \ cm}\\\sf{d_2 = 12 \ cm}\end{cases}$

Substituting values in the formula,

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$:\implies\sf Area_{(rhombus)} = \dfrac{1}{\cancel{ \: 2}} \times 9 \times \cancel{12} \\\\\\:\implies\sf Area_{(rhombus)} = 9 \times 6 \\\\\\:\implies\boxed{\frak {Area_{(rhombus)} = 54 \ cm^2}}$

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$\therefore\:\underline{\sf{Area \: of \: the \: rhombus \: is \: \bf{54 \: cm^2.}}}$

Answer 2

$\huge\bf{\underline{\underline{\gray{GIVEN:-}}}}$

• The diagonals of a rhombus are in ratio 3:4 .

• If the longer diagonal is 12cm .

$\huge\bf{\underline{\underline{\gray{TO\:FIND:-}}}}$

• The area of the rhombus .

$\huge\bf{\underline{\underline{\gray{SOLUTION:-}}}}$

Let,

• The smaller diagonal of the rhombus be '3a' .

• And the longer diagonal of the rhombus be '4a' .

☯︎ According to the question,

⇒ 4a = 12

⇒ a = $\rm{\dfrac{12}{4}}$

⇒ a = 3 cm

✞︎ Hence the smaller diagonal of the rhombus is

$\implies$ 3a

$\implies$ 3 × 3

$\implies$ 9 cm

☞︎︎︎ We know that,

$\purple\bigstar\:\bf{\red{\overbrace{\underbrace{\blue{Area\:of\:rhombus\:=\:\dfrac{1}{2}\times{d_1}\times{d_2}\:}}}}}$

Where,

• $\bf\red{d_1}$ = 9 cm

• $\bf\red{d_2}$ = 12 cm

⇛ Area of the rhombus = $\rm{\dfrac{1}{2}}$ × 9 × 12

⇛ Area of the rhombus = 9 × 6

⇛ Area of the rhombus = 54 cm²

$\huge\red\therefore$ The area of the rhombus is '54 cm²' .