## Wednesday, 5 October 2011

### Solutions for the Geometry Concept Problems-2 (Circle)

Q1) As the ÐAOB = ÐCOD, length of arc AB = length of arc CD

Q2) As the ÐAOC = ÐBOD, length of arc AC = length of arc BD

Q3) Given that ÐAOB is acute                  --> (1)
We know that ÐBOD = 1800ÐAOB    --> (2)
From (1) and (2) we can conclude that ÐBOD measures obtuse angle
and hence, ÐBOD > ÐAOC
and hence, Length of arc BD > Length of arc AB.

Q4) As the ÐAOB = ÐCOD,

Q5) As the ÐAOC = ÐBOD,

Q6) Given that ÐAOB = 600
Radius of the circle = 4 units
Length of arc AB = (angle made by the arc at the centre)*(2πr)/3600
= 60 * 2 * π * 4/360 = 4 π/3 units

Q7) Given that radius of the circle = r = 3 units
ÐAOB = 600
ÐCOD = ÐAOB = 600
= 600 + 600 = 1200
Total area of shaded region         = (total angle made by the arc at the centre)*(πr2)/3600
= 120 * π * 32 / 360 = 3 π Sq.Units

Q8) Given that, radius = r = 3    and    ÐAOB = 600
We know that AO = BO = r = 3
Consider ∆AOB, where in two sides AO and BO are equal in length
Hence angles opposite to these sides are equal,
ie., ÐOAB = ÐOBA
ÐOAB + ÐOBA + ÐAOB = 1800
ÐOAB + ÐOBA = 1800 - 600 = 1200
ÐOAB = ÐOBA = 1200/2 =600
As all the angles in ∆OAB are equal to 600, it is an equilateral triangle and
hence length of AB = length of OA =  length of OB = r =3 units

Q9)Let ÐAOB = α
Length of arc AB = α * 2 * π * r/360                                          --> (1)
Angle made by arc ACDB at centre = 360- α
Length of arc ACDB = (360-α) * 2 * π * r/360                        --> (2)
Given that, Length of arc ACDB = 2 * Length of arc AB     --> (3)
From (1), (2) and (3),
(360-α) * 2 * π * r/360 = 2 * α * 2 * π * r/360
(360-α) = 2 α => 360 = 3α => α = 1200