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If ABCD is a quadrilateral inscribed in a unit circle whose side AB=1 and diagonal BD=√3what is the value of AD

If ABCD is a quadrilateral inscribed in a unit circle whose side AB=1 and diagonal BD=√3what is the value of AD

Grade:12

2 Answers

Alicia Cotta
27 Points
7 years ago

Solution: 

 

AB = 1, BD = 3 , OA = OB = OD = 1

The given circle of radius 1 is also circumcircle of triangle ABD

R = 1 for triangle ABD

a/sinA = 2R

  3/sinA = 2R = 2

sinA = 3/2

A = 60°

and hence C = 120o

Also by cosine rule on triangle ABD,

(3)² = 1² + x² 2x cos60°

or   x² 2 = 0

x = 2

Now, area ABCD = triangle ABD + triangle BCD

Machine generated alternative text:

 

Machine generated alternative text: -g $cdl  2

cd = 1 , c²d² = 1

Also by cosine rule on triangle BCD we have

(3)² = c² + d² 2cd cos120°

= c² + d² + cd

c² + d² = 2 or cd = 1

 c² and  d² are the roots of t² 2t +1= 0

c² =  d² = 1

BC = 1 = CD and AD = x = 2

Alicia Cotta
27 Points
7 years ago

Solution: 

 

AB = 1, BD = √3 , OA = OB = OD = 1

The given circle of radius 1 is also circumcircle of triangle ABD

∴ R = 1 for triangle ABD

∴ a/sinA = 2R

∴  √3/sinA = 2R = 2

∴ sinA = √3/2

∴ A = 60°

and hence C = 120o

Also by cosine rule on triangle ABD,

(√3)² = 1² + x² −2x cos60°

or   x² −x −2 = 0

∴ x = 2

Now area ABCD = triangle ABD + triangle BCD

=3\sqrt{3}/4 –  ½ ( 1.2. sin60^{\circ} ) + ( ½ .c.d. sin120°)

3\sqrt{3}/4 – \sqrt{3}/2 + \sqrt{3}/4 c.d

cd = 1 , ⇒ c²d² = 1

Also by cosine rule on triangle BCD we have

(√3)² = c² + d² − 2cd cos120°

= c² + d² + cd

∴ c² + d² = 2 or cd = 1

∴ c² and  d² are the roots of t² − 2t +1= 0

∴ c² =  d² = 1

∴ BC = 1 = CD and AD = x = 2

 

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