Question

If one angle of a rhombus is 60° then find ratio of diagonals

Answer 1

Answer:

AC : BD = √3 AD : AD = √3 : 1

Step-by-step explanation:

Hey there!

We will use the following things in our solution :

#1 : All sides of a rhombus are equal

#2 : The diagonals bisect each other and are mutual perpendicular bisectors.

#3 : Relation between sides in a equilateral triangle and Right angled triangle.

First of all, A0 = AC/2 [ Diagonals bisect each other ]

Now, let's Consider that AD = AB [ A sides of a rhombus are equal ]

Also, One of the angle is given.

∠A = 60°

Now, Consider the triangle ΔABD,

We have AB = AD,

So It is an isosceles triangle.

So ∠B = ∠D

Now Use Angle Sum property;

∠A + ∠B + ∠C = 180

60 + 2∠B = 180

∠B = 120/2 = 60°

So , We found that,

∠A = ∠B = ∠C.

So, It is equilateral triangle.

From this , AB = AD = BD. [ Equation A ]

In ΔAOD,

∠AOD = 90° [ Diagonals are mutually perpendicular bisector ]

ΔAOD is right angled triangle.

So,

AD² = AO² + OD² [ Pythagoras theorem ]

OD = BD/2 = AD/2 [ AD = BD ]

AD² = ( AD/2)² + AO²

AD² - AD²/4 = AO²

3AD²/4 = AO²

AO = √3AD/2

AC = 2 * AO = 2 ( √3/2 AD) = √3 AD

From Equation A, BD = AD

Now, AC : BD = √3 AD : AD = √3 : 1