Question

The diagonals of a rhombus are in the ratio 3 : 4. If the longer diagonal is 12 cm, then find the area of

the rhombus​

Answer 1

Given :-

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

The longer diagonal is 12cm.

To Find :-

Area of rhombus ?

Solution :-

Let us assume that, diagonals of a rhombus are 4x and 3x respectively.

As,

→ 4x > 3x.

So ,

→ Longer diagonal = 12cm.

→ 4x = 12cm.

dividing both sides by 4,

→ x = 3 .

Therefore,

→ Longer diagonal = 12cm.

→ shorter diagonal = 3x = 3 * 3 = 9cm.

Hence,

→ Area of rhombus = (1/2) * Diagonal 1 * Diagonal 2.

→ Area = (1/2) * 12 * 9

→ Area = 6 * 9

→ Area = 54 cm². (Ans.)

Some Properties of Rhombus :-

• All sides of the rhombus are equal.
• The opposite sides of a rhombus are parallel.
• Opposite angles of a rhombus are equal.
• In a rhombus, diagonals bisecting each other at right angles.
• Diagonals bisect the angles of a rhombus.
• The sum of two adjacent angles is equal to 180 degrees.
• The two diagonals of a rhombus form four right angled triangles which are congruent to each other.
• You will get a rectangle when you join the mid point of the sides.
• You will get another rhombus when you join the mid points of half the diagonal.
• Around a rhombus, there can be no circumscribing circle.
• Within a rhombus, there can be no inscribing circle.

Answer 2

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.

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

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

$\begin{gathered}\\\end{gathered}$

$\sf \red{By \: using \: the \: formula,}$

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

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

$\sf \red{Substituting \: values \: in \: the \: formula,}$

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$\begin{gathered}\\\end{gathered}$

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

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