Question

a rhombus shaped field has green grass for 18 cows to graze if each side of the rhombus is 30 m and its longer diagonal is 48 former how much area of grass field with will it be getting​

Answer 1

Given :-

• The side of rhombus = 30m
• The diagonal of rhombus = 48 m

To Find :-

• The area of rhombus field = ?

Solution :-

• To calculate the area of rhombus field at first we have to find area ∆. Let assume ABCD is rhombus with longest diagonal BD = 48m. As we can notice in the given figure that two triangle ara formed ∆ABD adn ∆ BCD. Here we have to proof congruency of two triangles which is given in the figure. By Applying herons formula to calculate Area of triangle.

As per the given figure :-

⇒ Area of ∆ ABD = Area of ∆ BCD

• Let's proof by congruency:-

⇒ AB = CD (AB || CD)

⇒ BD = BD (Common side)

⇒ AD = BC (AD || BC)

∴ ∆ ABD ≅ ∆ BCD ( by S.S.S criteria of congruency)

∴ ∆ ABD = ∆ BCD ( by CPCTC)

• Now calculate Area pf ∆ BCD :-

⇒ Let , BC = a , CD = b. BD = C

• a = 30. b = 30. c = 48m

⇒ S = a + b + c/2

⇒ S = 30 + 30 + 48/2

⇒ S = 108/2 => 54m

• By applying herons formula :-

⇒ Area of ∆ BCD = √s(s - a)(s - b)(s - c)

⇒√54(54 - 30)(54 - 30)(54 - 48)

⇒√54(24 × 24 × 6)

⇒√ (3 × 3 × 3 × 2)(8 × 3)(8 × 3)(3 × 2)

⇒√(3 × 3)(3 × 3)(3 × 3)(8 × 8)(2 × 2)

⇒ 3 × 3 × 3 × 8 × 2

⇒ 432m²

• Now calculate area of rhombus

⇒ Area of rhombus = ∆ ABD + ∆ BCD

⇒Area of rhombus = 432 + 432

⇒Area of rhombus = 864 m²

Hence,

• The area of rhombus field = 864 m²

Answer 2

Appropriate Question:-

a rhombus shaped field has green grass for 18 cows to graze if each side of the rhombus is 30 m and its longer diagonal is 48m how much area of grass field each cow will get?

Solution:-

Figure:-

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Let The rhombus be ABCD

We know sides of a rhombus are equal

$\\ \sf\longmapsto AB=BC=CD=AD=30m$

• Diagonal=AC=48m

Now

$\sf In\:\Delta ABC,$

$\boxed{\sf Semiperimeter(s)=\dfrac{a+b+c}{2}}$

$\\ \sf\longmapsto s=\dfrac{30+30+48}{2}$

$\\ \sf\longmapsto s=\dfrac{108}{2}$

$\\ \sf\longmapsto s=54m$

Using Herons Formula

$\boxed{\sf Area_{(Triangle)}=\sqrt{s(s-a)(s-b)(s-c)}}$

$\\ \sf\longmapsto Area_{(\Delta ABC)}=\sqrt{54(54-30)(54-30)(54-48)}$

$\\ \sf\longmapsto Area_{(\Delta ABC)}=\sqrt{54(24\times 24\times 6)}$

$\\ \sf\longmapsto Area_{(\Delta ABC)}=\sqrt{54\times 3456}$

$\\ \sf\longmapsto Area_{(\Delta ABC)}=\sqrt{186,624}$

$\\ \bf\longmapsto Area_{(\Delta ABC)}=432m^2$

As $\sf \Delta ABC=\Delta ADC$

$\\ \bf\longmapsto Area_{(\Delta ADC)}=432m^2$

Now

$\boxed{\sf Area_{(Rhombus)}=Area_{(\Delta ABC)}+Area_{(\Delta ADC)}}$

$\\ \sf\longmapsto Area_{(Rhombus)}=432+432$

$\\ \sf\longmapsto Area_{(Rhombus)}=864m^2$

$\\ {\therefore{\underline{\underline{\footnotesize{\sf {Area\:of\:the\:grass\:field\:is\:864m^2.}}}}}}$

• Number of cows=18

Each cow will get:-

$\\ \sf\longmapsto \dfrac{Area\:of\:Field}{No\:of\:cows}$

$\\ \sf\longmapsto \dfrac{864}{18}$

$\\ \sf\longmapsto 48m^2$

$\\ {\therefore{\underline{\underline{\footnotesize{\sf Each\:cow\:will\:get\:48m^2\:grass\:field\:to\:eat.}}}}}$