Class 9 Maths Question Answers - Circles

Q1. In the figure, if ∠ ACB = 35°, then find the measure of ∠ OAB.

Solution: ∠AOB = 180° -  (∠1 + ∠2)
= 180° - (35° + 35°)                     [∵ AO = OB ∴ ∠1 = ∠2]
= 110°
∠ACB = (1/2)(∠AOB) = (1/2) (110°) = 55°

Q2. If points A, B and C are such that AB ⊥ BC and AB = 12 cm, BC = 16 cm. Find the radius of the circle passing through the points A, B and C.
 Solution: ∵ AB ⊥ BC
∴ ∠B = 90°                  [Angle in a semicircle]
⇒ AC is a diameter 

AC = 20

Radius = 10cm

Q3. The angles subtended by a chord at any two points of a circle are equal. Write true or false for the above statement and justify your answer.
 Solution: False. If two points lie in the same segment only, then the angles will be equal otherwise they are not equal.

Q4. Two chords of a circle of length 10 cm and 8 cm are at the distance 8.0 cm and 3.5 cm, respectively from the centre, state true and false for the above statement.
 Solution: False. As the larger chord is at a smaller distance from the centre.

Q4. If AB = 12 cm, BC = 16 cm and AB is perpendicular to BC, then the radius of the circle passing through the points A, B and C is 

Solution: 

Assuming that AB = 12 cm and BC = 16 cm

If BC and AB are in a circle, then AC will be the circle's diameter.

[circular's diameter forms a right angle to the circle]

Apply the Pythagorean theorem to the ABC right angle.

AC2=AB2+BC2⇒

AC2=(12)2+(16)2⇒

AC2=144+256⇒ AC2=400

⇒ AC2=√400=20 cm

[using the square root of a positive number since the diameter is always positive]

∴ Circle radius = 1/2(AC)=1/220=10 cm

As a result, the circle's radius is 10 cm.

Q5. In Figure, if ∠ABC = 20º, then ∠AOC is equal to:  

Solution: Assumed: ABC = 20°...(1)

We are aware that an arc's angle at the circle's center is twice as large as its angle at the rest of the circle.

= ∠AOC = 2(20°) (From (1))

=  ∠AOC = 40°

∠AOC thus equals 40°

Q6. Two chords AB and AC of a circle with center O are on the opposite sides of OA. Then ∠OAB = ∠OAC. 

Write true or false for the above statement and justify your answer.

Q9.  Two chords AB and CD of a circle are each at distances 4 cm from the center. Then AB = CD.

Write true or false for the above statement and justify your answer. 

Solution: The right answer is True.

The above assertion is accurate since chords that are equally spaced from a circle's center have equal lengths.

AB=CD 

Q10.  If A, B, C, and D are four points such that ∠BAC = 30° and ∠BDC = 60°, then D is the center of the circle through A, B and C. 

Write true or false for the above statement and justify your answer. 

Solution: 

False

Because there are numerous places D where ∠BDC=60°, none of which can be the center of the circle formed by points A, B, and C.