Question

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

Answer 1

$\huge\bold{S O L U T I O N }$

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