9th Grade > Mathematics
CIRCLES MCQs
:
D
Circle is the set of all the points in a plane which are at a given distance from a fixed point in the plane. That fixed point is known as the centre of the circle.
:
B
A quadrilateral is called cyclic if all the four vertices of it lie on a circle and the sum of its opposite angles is 180∘.
:
C
In the figure, ∠ABD=50∘
Since the angles subtended by an arc in the same segment are equal, we have ∠ABD=∠ACD.
⇒X=∠ACD=50∘
:
B
The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
Given∠ABC=50∘,
∴∠AOC=2 ∠ABC=2×50∘=100∘
Also, since OA and OC are radii of the circle, OA=OC.
⟹∠OAC=∠OCA
In △AOC,
x+x+∠AOC=180∘.
⟹2x=180∘−∠AOC=180∘−100∘=80∘
⟹x=40∘
:
Chord is a line segment that joins two points on the circumference of a circle. Diameter is the longest chord of a circle which passes through centre joining the two points on the circumference of a circle.
:
C
A quadrilateral is cyclic if its opposite angles are supplementary.
In square, rectangle and isosceles trapezium, the opposite angles are supplementary, that is, the sum of the opposite angles is 180∘.
In a parallelogram, the opposite angles are equal and may not add to 180∘. Hence, a parallelogram cannot be a cyclic quadrilateral.
:
A
The line drawn through the centre of a circle perpendicularly bisects a chord.
Hence, ∠x=90∘.
:
A and B
It can be observed that ∠AOC=∠AOB+∠BOC=60∘+30∘=90∘.
We know that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
So,∠ADC=12∠AOC=12×90∘=45∘
∴∠ADC=45∘
:
A
Given that ∠BAC=40∘ and ∠DBC=80∘.
Since the angles formed by the same segment are equal,
∠BDC=∠BAC=40∘.
In ΔBDC,
∠BDC+∠DBC+∠BCD=180∘. [Angle sum property]
i.e., 40∘+80∘+∠BCD=180∘
⟹∠BCD=180∘−120∘=60∘
:
C
In ΔOAB,
AB = OA = OB = radius of the circle.
∴ΔOAB is an equilateral triangle.
Therefore, each interior angle of this triangle will be equal to 60∘.
∴∠AOB=60∘.
Since the angle subtended by an arc of the circle at its centre is double the angle subtended by it at any point on the remaining part of the circle, we have
∠ACB=12∠AOB=12×60∘=30∘.
Now in the cyclic quadrilateral ACBD,
∠ACB+∠ADB=180∘. [Opposite angles in a cyclic quadrilateral are supplementary]
∴∠ADB=180∘−∠ACB=180∘−30∘=150∘
Therefore, the angles subtended by the chord AB at a point on the major arc and the minor arc are 30∘ and 150∘ respectively.