ABCD is a rhombus in which angle A =60° .find ratio AC:BD
Answers
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
Read more on Brainly.in - https://brainly.in/question/2618565#readmoreHey 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
Read more on Brainly.in - https://brainly.in/question/2618565#readmoreHey 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
Read more on Brainly.in - https://brainly.in/question/2618565#readmoreHey 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
Read more on Brainly.in - https://brainly.in/question/2618565#readmore