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Directions : Read the following questions carefully and choose the right answer.
1
PQ is a chord of a circle with centre O and SOR is a line segment originating from a point S on the circle and intersecting PQ produced at R such that QR = OS. If ∠QRO = 30° then ∠POS = ?
» Explain it
C



Let radius be 'r' and ∠POS = x° 
 
ΔOQR isoceles ∴∠QOR = 30°
 
∴ ∠OQR = 120° (Sum of all angles of ΔOQR = 180°)
 
∴ ∠OQP = 60° (Supplementary agnle) 
 
ΔOPQ isoceles since OP = OQ = r
 
∴ ∠OQP = 60° = ∠OQP
 
∴ ∠ POQ = 60° = [Sum of all angle of Δ = 180° ] 
 
Now SOR is a straight line
 
∴ x + 60° + 30° = 180°
 
∴ x = 90° 

Hence, option (C) is  correct.

2
O and O' are repectively the orthocentre and cicumcentre of an acute angled triangle PQR. the point P and O are joined and produced to meet the side QR at S. If ∠PQS = 60° and ∠QO'R = 130° then ∠RPS = ?
» Explain it
B



∠ PQS = 60°

∠QO'R = 130°

∠ QPR =  1  × 130° = 65°
2

⇒ ∠ QRP = 180° – 60° – 65° = 55°

⇒ ∠PO'Q = 110° 

In Δ QO'R

QO' = O'R

⇒ ∠O'QR = ∠O'RQ = 25° 

∵ ∠O'QR + ∠O'RQ = 50°

⇒ ∠PQO' + ∠QPO' = 35° 

∵ ∠PQO' + ∠QPO' = 70°

Similarly, ∠O'PR = 30°

∴ ∠RPS = 35°

Hence, option (B) is correct.

3
In the given figure below, ∠AOB = 48° and AC and OB intersect each other at right angles. What is the measure of ∠OBC? (O is the centre of the circle)

» Explain it
B
∠AOB = 48°

So, ∠ACB =  1 ∠AOB
2

=  1  × 48° = 24°
2

(As angles made by same arc AB)

Given AC and OB intersect each other at right angle.

∠ CQB = 90°

∠ CBQ = 180° – (90° + 24°) = 66°

so , ∠ OBC = 66°

Hence, option B is correct.


4
In a right angled triangle, the circumcentre of the triangle lies.
» Explain it
C


∠APB = 90°

AB = Diameter = Hypotenous of triangle APB

As, the angle of semicircle is right angle

so, the circumcentre lies on midpoint of hypoteneous

Hence, option (C) is correct.

5
AB is the diameter of a circle with centre O and radius OD is perpendicular to AB. Find the angle BAD
» Explain it
B
In ∆ AOD :
 
OA = OD (radius)
 
∠ AOD = 90 ( as OD is perpendicular to AB )
 
So ∆ AOD is isosceles having OA and OD sides equal and one angle as 90
 
So the remaining wo angles are 45 each
 
Hence ∠ BAD = 45°

Therefore, option (B) is correct.