The length of each side of a rhombus is 10 and the measure of an angle of the rhombus is 60. Find the length of the longer diagonal of the rhombus.

the diagonals of a rhombus are perpendicular bisectors of each other, and are also angle bisectors. The 4 congruent right triangles the rhombus is cut into by the two diagonals are 30-60-90 degree triangles, the longer leg is √3/2 of the hypotenuse, √3/2 of 10=5√3. so the longer diagonal is twice of 5√3=10√3

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AnkitM AnkitM · Answer 2 · 2022-05-25T03:44:20+02:00

The length of the longer diagonal of the rhombus is 10[tex]\sqrt{3}[/tex].

What is a rhombus?

'In plane Euclidean geometry, a rhombus is a quadrilateral whose four sides all have the same length.'

According to the given problem,

Length of each side of the rhombus = 10

Measure of angle = 60°

In the right angled triangle AOD,

∠OAD = 30°

∠AOD = 90°

AD = 10

⇒ cos 30 = [tex]\frac{Base}{Hypotenuse}[/tex]

⇒ cos 30 = [tex]\frac{OA}{10}[/tex]

⇒ 10 × [tex]\frac{\sqrt{3} }{2}[/tex] = OA

⇒ 5[tex]\sqrt{3}[/tex] = OA

We know, in the rhombus ADCB,

Longer diagonal, AC = 2 × AO

Therefore,

Longer diagonal = 2 × 5[tex]\sqrt{3}[/tex]

= 10[tex]\sqrt{3}[/tex]

Hence, we can conclude, the length of the longer diagonal is 10[tex]\sqrt{3}[/tex].

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