Plane Geometry
- If the length of a chord of a circle is 16 cm and is at a distance of 15 cm from the centre of the circle, then the radius of the circle (in cm) is :
-
View Hint View Answer Discuss in Forum
According to question , we draw a figure of a circle with centre O ,
Here , AB = chord = 16 cm.
OC ⊥ AB
∴ AC = CB = 8 cm.
OC = 15 cm.
∴ OA = √OC² + CA²Correct Option: C
According to question , we draw a figure of a circle with centre O ,
Here , AB = chord = 16 cm.
OC ⊥ AB
∴ AC = CB = 8 cm.
OC = 15 cm.
∴ OA = √OC² + CA²
OA = √15² + 8²
OA = √225 + 64 = √289 = 17 cm.
- Two circles touch each other internally. The greater circle has its radius as 6 cm and the distance between the centres of the circles is 2 cm. The radius of the other circle is
-
View Hint View Answer Discuss in Forum
As per the given in question , we draw a figure of two circles touch each other internally ,
Correct Option: B
As per the given in question , we draw a figure of two circles touch each other internally ,
Given , OA = 6 cm.
OO' = 2 cm.
∴ O'A = OA – OO' = 6 – 2 = 4 cm.
- AB is a chord of a circle with 0 as centre. C is a point on the circle such that OC ⊥ AB and OC intersects AB at P. If PC = 2 cm and AB = 6 cm then the diameter of the circle is
-
View Hint View Answer Discuss in Forum
On the basis of question we draw a figure of a circle with centre O ,
OC ⊥ AB
∴ AP = PB = 3 cm
PC = 2 cm
If OA = OC = r cm
then, OP = (r – 2) cm.
From ∠ OAP,
OA² = AP² + OP²
⇒ r² = 3² + (r – 2)²
⇒ r² – (r – 2)² = 9
⇒ r² – r² + 4r – 4 = 9⇒ 4r = 13 ⇒ r = 13 = cm. 4
Correct Option: B
On the basis of question we draw a figure of a circle with centre O ,
OC ⊥ AB
∴ AP = PB = 3 cm
PC = 2 cm
If OA = OC = r cm
then, OP = (r – 2) cm.
From ∠ OAP,
OA² = AP² + OP²
⇒ r² = 3² + (r – 2)²
⇒ r² – (r – 2)² = 9
⇒ r² – r² + 4r – 4 = 9⇒ 4r = 13 ⇒ r = 13 = cm. 4 ∴ Diameter of circle = 2 × 13 = 13 cm. = 6.5 cm. 4 2
- Chord PQ is the perpendicular bisector of radius OA of circle with centre O (A is a point on the edge of the circle). If the length of Arc PAQ = 2π/3 . What is the length of chord PQ ?
-
View Hint View Answer Discuss in Forum
As per the given in question , we draw a figure of a circle with centre O,
PQ is perpendicular bisector of OA.
∴ OP = OQ = PA = AQ
∴ OPAQ is a rhombus.
As we know that the angle sutended at the centre by an arc is twice to that at the circumference
∴ 2 ∠ PAQ = Reflex ∠POQ
⇒ 2 ∠ PAQ = 360° – ∠POQ
⇒ 3∠ PAQ = 360°
(∵ ∠PAQ = ∠POQ)⇒ ∠PAQ = 120° = ∠POQ = 2π 3 Again, radius (r) = l = 2π/3 = 1 θ 2π/3
∴ From ∆ OPB
OP = 1 unit
∠POB = 60°∴ sin 60° = PB OP
Correct Option: B
As per the given in question , we draw a figure of a circle with centre O,
PQ is perpendicular bisector of OA.
∴ OP = OQ = PA = AQ
∴ OPAQ is a rhombus.
As we know that the angle sutended at the centre by an arc is twice to that at the circumference
∴ 2 ∠ PAQ = Reflex ∠POQ
⇒ 2 ∠ PAQ = 360° – ∠POQ
⇒ 3∠ PAQ = 360°
(∵ ∠PAQ = ∠POQ)⇒ ∠PAQ = 120° = ∠POQ = 2π 3 Again, radius (r) = l = 2π/3 = 1 θ 2π/3
∴ From ∆ OPB
OP = 1 unit
∠POB = 60°∴ sin 60° = PB OP ⇒ PB = √3 2 ∴ PQ = 2 × √3 = √3 unit 2
- In the adjoining figure ∠AOC = 140° where O is the centre of the circle then ∠ABC is equal to :
-
View Hint View Answer Discuss in Forum
According to question , we draw a figure of a circle with centre O
Given that , ∠AOC = 140°
Angle subtended by arc AC at the centre = Reflex angle AOC = 360° – 140° = 220°Correct Option: A
According to question , we draw a figure of a circle with centre O
Given that , ∠AOC = 140°
Angle subtended by arc AC at the centre = Reflex angle AOC = 360° – 140° = 220°∴ ∠ABC = Angle at the circumference = 220 = 110° 2
