Question

$\huge \bf Question$

A rhombus of side of 10 cm has two angles 60° each. Find the length of diagonals and also find its area

All the Best :)

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Answer 1

Answer :-

Let ABCD be a rhombus of side 10 cm and O is the intersection point of diagnols.

$\sf \angle BAD = \angle BCD = 60 \degree$

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According to the properties of rhombus, it bisects the diagnol into two equal parts :-

→ AO = OC

→ BO = OD

In the right triangle AOB :-

∠ AOB = 30°

cos 30° = OA/OB

→ √3/2 = OA/10

→ OA = 5√3

OA is half of the diagonal AC -

→ AC = 2 ( OB )

→ AC = 10 √3 cm

sin30° = OB/AB

→ 1/2 = OB/10

→ OB = 5 cm

OB is half of the diagonal BD -

→ BD = 2 ( OB )

→ BD = 10 cm

Diagnols of rhombus =

• BD = 10 cm
• BD = 10 cm AC = 10√3 cm

Calculating Area :-

Area of rhombus = 1/2 × Product of diagnols

= 1/2 × 10 × 10√3 cm²

= 50√3 cm²

$\boxed{\rm Area\: of\: rhombus = 50\sqrt{3} cm^2 }$

Trignometric table :-

$\bullet\:\sf Trigonometric\:Values :\\\\\boxed{\begin{tabular}{c|c|c|c|c|c}Radians/Angle & 0 & 30 & 45 & 60 & 90\\\cline{1-6}Sin \theta & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} & 1\\\cline{1-6}Cos \theta & 1 & \dfrac{\sqrt{3}}{2}& \dfrac{1}{\sqrt{2}}& \dfrac{1}{2}&0\\\cline{1-6}Tan \theta&0& \dfrac{1}{\sqrt{3}}&1&\sqrt{3}&Not D{e}fined\end{tabular}}$

Answer 2

Given :-

A rhombus of side of 10 cm has two angles 60° each.

To find :-

• Length of diagonals
• Area of Rhombus

Explanation :-

★ Properties of Rhombus ★

• The diagonals of a rhombus bisect each other at 90°

• A pair of adjacent sides are equal

Solution :-

• WXYZ is a Rhombus
• ∠XYZ = 60°
• ∠XWZ = 60°
• WX = 10 cm
• ∠XWO = 30°
• WO = OY and XO = OZ

In ∆WOX

→ sin 30° = OX/WX

→ ½ = OX/10

→ 2 × OX = 10

→ OX = 10/2 = 5cm

Now,

→ cos 30° = WO/WX

→ √3/2 = WO/10

→ √3 × 10 = 2 × WO

→ 10√3 = 2 × WO

→ WO = 10√3/2 = 5√3 cm

Hence,

• Required diagonals
• XZ = 2OX = 2 × 5 = 10cm
• WY = 2WO = 2 × 5√3 = 10√3cm

Area of rhombus

→ ½ × product of diagonals

→ ½ × XZ × WY

→ ½ × 10 × 10√3

→ 5 × 10√3

→ 50√3 cm²

→ 50 × 1.73

→ 86.5 cm²

•°• Area of rhombus is 86.5 cm²