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ABCD is a rhombus in which ∠A = 60°. Find the ratio AC : BD.

Rectilinear Figures

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Answer

Rhombus ABCD is shown in the figure below:

ABCD is a rhombus in which ∠A = 60°. Find the ratio AC : BD. Rectilinear Figures, ML Aggarwal Understanding Mathematics Solutions ICSE Class 9.

In △ABD,

⇒ AB = AD (Sides of rhombus are equal.)

⇒ ∠B = ∠D = x (let) (∵ angles opposite to equal sides are equal)

⇒ ∠A + ∠B + ∠D = 180°

⇒ 60° + x + x = 180°

⇒ 2x = 180° - 60°

⇒ 2x = 120°

⇒ x = 60°.

∴ ABD is an equilateral triangle.

So, BD = AB = AD = a (let)

Since, diagonals of rhombus bisect each other,

OB = a2\dfrac{a}{2}

In right angled triangle AOB,

AB2 = AO2 + OB2

a2 = AO2 + (a2)2\Big(\dfrac{a}{2}\Big)^2

AO2 = a2 - a24=3a24\dfrac{a^2}{4} = \dfrac{3a^2}{4}

AO = 3a2\dfrac{\sqrt{3}a}{2}

AC = 2AO = 3a\sqrt{3}a

AC : BD = 3a:a=3:1\sqrt{3}a : a = \sqrt{3} : 1.

Hence, AC : BD = 3:1\sqrt{3} : 1.

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