Question

Perimeter of a rhombus is 2p units. sum of their diagonal is m unit. what is the area of rhombus

Answer 1

The area of the rhombus is ¼ * (m² − p²).

Step-by-step explanation:

Step 1:

The perimeter of the rhombus = 2p units  

We have,  

Perimeter = 4 * side of the rhombus

⇒ 2p = 4 * side of the rhombus

⇒ Side of the rhombus = p/2 units  

Step 2:

Let’s assume the diagonal, AC = 2y and BD = 2x

∴ OA = OC = y and OB = OD = x

We know that the diagonals of a rhombus bisect at 90 deg

∴ ∠AOB = ∠BOC = ∠COD = ∠AOD = 90°

Consider right-angled △OBC and applying Pythagoras theorem, we get

OB² + OC² = BC²

⇒ x² + y² = p²/4

⇒ 4x² + 4y² = p² ..... (i)

Also given that the sum of the diagonals is m units i.e.,  

2x + 2y = m

Squaring throughout

⇒ 4x² + 2y² + 8xy = m²

⇒ 4x² + 4y² = m² – 8xy ..... (ii)

Step 3:

Now, from (i) and (ii)

m²– 8xy =p²

⇒ 8xy = m² − p²

⇒ 4 × (2xy) = m² − p²

⇒ 2xy = ¼ * (m² − p²) ….. (iii)

Thus,

The area of the Rhombus is given by,

= ½ * diagonal 1 * diagonal 2

= ½ * AC * BD

= ½ * 2x * 2y

= ½ * 4xy

= 2xy

Substituting from (iii)

= ¼ * (m² − p²)

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Also View:

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Answer 2

The area of the rhombus is 1$\frac{1}{4} (m^{2}-p^{2} )$

• Perimeter of the rhombus= 2p units

                   4 x sides = 2p

                      side= p/2

• Now, In rhombus ABCD

                     diagonal AC;  OA=OC=a ;AC=2a

                     diagonal BD;  OB=OD=b ;BD=2b

                                       a + b =p/2

                                       $a^{2} +b^{2} =\frac{p^{2} }{4}$

• On squaring

                                      $4a^{2}+4b^{2}=p^{2}$

•        But the sum of the diagonal is 2a+2b=m

•                              On squaring

                    $4a^{2}+4b^{2}+8ab=m^{2}$

                          $p^{2}+8ab=m^{2}$  

                          $8ab=m^{2}-p^{2}$   ; $4(2ab)=m^{2}-p^{2}$    

• we know that, area of the rhombus=$\frac{1}{2} d_{1} d_{2}$

                    $=\frac{1}{2} 2a 2b$

                    =  2ab

                    =    $\frac{1}{4} m^{2}-p^{2}$