Question

One of the angles of a rhombus with side 20cm is 60˚ ,then the lengths of the diagonals of this rhombus is__________ *
20 cm & 10 cm
20 cm & 20√3 cm
20√2 cm & 10√3 cm
10 cm & 10√3 cm​

Answer 1

Given :

• One of the angles of a rhombus with side 20 cm is 60˚

To find :

The lengths of the diagonals of this rhombus.

Solution :

• Properties of rhombus

★ All adjacent sides are equal

★ Diagonals are perpendicular bisector

★ Opposite sides are parallel

★ Opposite angles are equal

• According to the given properties

AD = DC = CB = AB = 20cm

°•°∠BAD = 60°

•°• ∠BAE = ∠DAE = 30°

Now,

• In ∆AED

→ sin 30° = DE/DA

→ 1/2 = DE/20

→ 2DE = 20

•°• DE = 20/2 = 10cm

• Again in ∆AED

→ cos 30° = AE/DA

→ √3/2 = AE/20

→ 20√3 = 2AE

→ AE = 20√3/2

•°• AE = 10√3cm

Length of diagonals of rhombus

→ AE = 10√3cm

→ AC = 2 × 10√3 = 20√3cm

→ DE = 10cm

→ DB = 2 × 10 = 20cm

•°• Length of diagonals are 20√3cm & 20cm

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Answer 2

Answer:

Given :-

• One of the angles of a rhombus with side 20 cm is 60°.

To Find :-

• What is the length of the diagonals of the rhombus.

Solution :-

$\longmapsto$ In right angled triangle ∆AED :

↦ $\sf sin 30^{\circ} =\: \dfrac{DE}{DA}$

As we know that, [ sin 30° = ½ ]

↦ $\sf \dfrac{1}{2} =\: \dfrac{DE}{20}$

By doing cross multiplication we get :

↦ $\sf 2(DE) =\: 20$

↦ $\sf 2DE =\: 20$

↦ $\sf DE =\: \dfrac{\cancel{20}}{\cancel{2}}$

➠ $\sf\bold{\green{DE =\: 10\: cm}}$

Now, again,

$\longmapsto$ In right angled triangle ∆AED :

↦ $\sf cos 30^{\circ} =\: \dfrac{AE}{DA}$

As we know that, [cos 30° = √3/2 ]

↦ $\sf \dfrac{\sqrt{3}}{2} =\: \dfrac{AE}{20}$

By doing cross multiplication we get :

↦ $\sf 2(AE) =\: 20\sqrt{3}$

↦ $\sf 2AE =\: 20\sqrt{3}$

↦ $\sf AE =\: \dfrac{\cancel{20}\sqrt{3}}{\cancel{2}}$

➠ $\sf\bold{\green{AE =\: 10\sqrt{3}}}$

Hence, the length of the diagonal are :

➲ $\sf DE =\: 10\: cm$

↦ $\sf DB =\: 2 \times 10$

➠ $\sf\bold{\red{DB =\: 20\: cm}}$

And,

➲ $\sf AE =\: 10\sqrt{10}$

↦ $\sf AC =\: 2 \times 10\sqrt{3}$

➠ $\sf\bold{\red{AC =\: 20\sqrt{3}}}$

$\therefore$ The length of the rhombus is 20 cm and 20√3 cm.

Hence, the correct options is option no (2) 20 cm and 20√3 cm .