Question

area of a rhombus is 216 cm ² and the diagonal is 24 cm. find the length of the other diagonal also find the side of the rhombus​

Answer 1

★ Concept

The above question is all based on rhombus and its properties. Understand the question first, we are provided with the area of rhombus 216 cm²; measure for the length of one diagonal 24 cm. Now, what's asked us to find is the length of the other diagonal and the side of the rhombus. For calculating the length of other diagonal, use the formula for Area of Rhombus i.e 1/2(D₁ × D₂) and since we have to calculate D₂, we'll simply substitute the other values i.e Area and D₁ in the formula. Secondly, In order to calculate the side of rhombus we'll make a use of Pythagoras Theorem.

Let's proceed with Calculation !!

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➤ Area of rhombus = 216 cm²

➤ Length of one of its diagonal (D₁) = 24 cm

$\underline{ \boxed{ \red{ \sf Area \: of \: Rhombus = \sf \dfrac{1}{2}( D_1 \times D_2)}}}$

On substituting all known values in the above formula.

$\sf216 = \dfrac{1}{2} (24 \times D_2)$

$\sf216 \times 2 = 24 \times D_2$

$\sf \cancel\dfrac{432}{24} = \bf D_2 \qquad \underline{\boxed{\purple{ \sf D_2 = 18 \: cm}}}$

For calculating the side of rhombus, must note :

• All four sides of Rhombus are equal.
• Diagonals bisects each other at 90°.

Using Pythagoras Theorem

$\sf Measure \: for \: Side = \sqrt{ {(12)}^{2} + {(9)}^{2} }$

$\sf Required \: Side = \sqrt{144 + 81} = \sqrt{225}$

$\sf \: Required \: Side = \pm\sqrt{225} = \red{ 15 \: cm}$

[Neglect the '-ve' sign for a side can never be negative]

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Additional Information

Rhombus

A rhombus is a special case of a parallelogram, and it is a four-sided quadrilateral. A square is a type of rhombus.

Below are provided some properties of Rhombus :-

1) The sum of interior angles of rhombus add up to 360 degrees.

2) The opposite angles of a rhombus are equal to each other.

3) The adjacent angle are supplementary.

4) In a rhombus, diagonals bisect each other at right angles.

5) All sides of the rhombus are equal.

6) Around a rhombus, there can be no circumscribing circle.

Answer 2

★ Given:-

Area of the rhombus is 216 cm2

Length of a diagonal = 18 cm

★ Concept used:-

The two diagonals bisect each other at 90° in a rhombus.

★ Formula used:-

$\sf :⇒ Area \: of \: rhombus = \frac{1}{2} \times product \: of \: two \: diagonal \:$

★ Solution :-

$:⇒\sf Area \: of \: rhombus = \frac{1}{2} \times d1 \times d2$

$\sf \: :⇒ 216 = \frac{1}{2} \times 24 \times d2$

$\sf \: :⇒ 216 = 12 \times d2$

$\sf \: :⇒ \frac{216}{12} = d2$

$\sf \fbox\red{ :⇒ d2 = 18}$

$\sf \: :⇒ so \: second \: diagonal \: = 18cm$

$:⇒\sf \: If \: the \: two \: diagonals \: of \: rhombus \\ \sf ABCD \: intersect \: at \: (O) \: then$

$\sf \: :⇒area \: of \: rhombus \: = a$

$\sf : ⇒ AO^{2} + BO^{2} = AB ^{2}$

$\sf \: : ⇒ ( \frac{18}{2} {)}^{2} + (\frac{24}{2} {)}^{2} = {a}^{2}$

$\sf \: : ⇒ 81 + 144 = a ^{2}$

$\sf \: : ⇒ a^{2} = 225$

$\sf\fbox\red{ :⇒ a = 15 cm}$

★ Answer :-

$\sf \: :⇒ {d}^{2} = 18 \\ \sf:⇒ side \: = 15$

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