Mathematics for IGCSE & O level - Circle Theorems (Section 2)
1
What is the property of angle subtended at the center by an arc?
Answer
(C)
It is twice the angle at the circumference subtended by the same arc.
2
What relationship exists between opposite angles of a cyclic quadrilateral?
Answer
(B)
They are supplementary.
3
What is the relationship between the lengths of the tangents drawn from an external point to a circle?
Answer
(A)
They are equal.
4
In a circle, a chord is drawn. A line is drawn from the center of the circle perpendicular to the chord. What can be said about the chord and the line?
Answer
(A)
The line bisects the chord and forms a right angle.
5
If a circle has a radius of 5 cm, and a chord is drawn 3 cm from the center, what is the length of the chord?
Answer
(C)
8 cm
6
In a cyclic quadrilateral, angles A and C are opposite angles. If angle A = 75 degrees, what is the measure of angle C?
Answer
(B)
105 degrees
7
In a circle with center O, chords AB and CD intersect at point E inside the circle. Which of the following are true statements regarding the angles formed?
Answer
A
C
D
8
What is the relationship between the lengths of the tangents from an external point to a circle?
Answer
(A)
Equal
9
What type of angle is formed in a semicircle?
Answer
(C)
A right angle
10
Which of the following statements about angles in circles are correct?
Answer
A
C
11
What angle is formed by a tangent and a radius at the point of tangency?
Answer
(C)
90 degrees
12
In a circle, a chord forms an angle at the centre of 100°. What is the angle subtended by the same chord at the circumference?
Answer
(C)
50°
13
If two tangents from a point P are drawn to a circle with center O, and angle between the tangents is 60 degrees, what is the angle between the tangents and the radii at the point of contact?
Answer
(C)
90 degrees
14
Which of the following is a key characteristic of a cyclic quadrilateral?
Answer
(C)
Opposite angles are supplementary.
15
Which of the following is the converse of the theorem stating 'Equal chords are equidistant from the center'?
Answer
(A)
Chords equidistant from the center are equal.
16
What relationship exists between the angles marked with letters in the circles?
Answer
A
B
C
17
What type of angle is inscribed in a semicircle?
Answer
(C)
Right
18
If two tangents are drawn from a point outside a circle, what is always true about their lengths?
Answer
(C)
They are equal in length.
19
Which property relates to the lengths of tangents drawn from a point outside the circle?
Answer
(A)
They are equal.
20
If two circles touch externally, how many common tangents can be drawn?
Answer
(C)
3
21
How does the angle at the center of a circle, subtended by an arc, relate to the angle at the circumference subtended by the same arc?
Answer
(C)
The angle at the center is twice the angle at the circumference.
22
If a tangent touches a circle at point P and the radius is OP, what is the angle between them?
Answer
(C)
90°
23
Two tangents are drawn from an external point to a circle. What is true about the lengths of these tangents?
Answer
(B)
They are equal in length.
24
Which statement correctly describes the property of angles in the same segment of a circle?
Answer
(C)
They are equal.
25
If a chord of a circle subtends an angle of 70 degrees at the center, what angle does it subtend at any point on the major arc?
Answer
(A)
35 degrees
26
In a circle, if two chords are the same length, how are they positioned relative to the center?
Answer
(A)
They are equidistant from the center.
27
If two angles in the same segment of a circle are subtended by the same arc, how are these angles related?
Answer
(C)
They are equal.
28
What is the relationship between the angles in the same segment of a circle?
Answer
(C)
They are equal.
29
What can be determined about the quadrilateral formed by two radii and two tangents from an external point to a circle?
Answer
(C)
It's always a kite.
30
Which of the following must be true in a circle?
Answer
A
B
C
31
What is the relationship between the angle at the center of a circle and the angle at the circumference subtended by the same arc?
Answer
(B)
The angle at the center is twice the angle at the circumference.
32
What is the relationship between a tangent to a circle and the radius drawn to the point of contact?
Answer
(B)
They are perpendicular.
33
What can you deduce about the chords which are equidistant from the centre of a circle?
Answer
(A)
The chords are of equal length.
34
If a line segment from the center of a circle bisects a chord (but isn't a radius), what's the relationship between the line segment and the chord?
Answer
(B)
They are perpendicular.
35
What can be concluded if two circles touch externally?
Answer
(C)
The distance between their centers is the sum of their radii.
36
What can you say about the triangles formed when two tangents intersect at a point outside the circle, where the radii meet the tangents?
Answer
(B)
They are congruent.
37
What is the necessary and sufficient condition for a quadrilateral to be cyclic?
Answer
(A)
Its opposite angles are supplementary.
38
What is the correct relationship between an angle formed by two intersecting chords inside a circle and the arcs they intercept?
Answer
(C)
The angle is half the sum of intercepted arcs.
39
In a circle, if a chord is drawn to the circle, which of the following is correct?
Answer
(A)
The perpendicular from the centre to the chord bisects the chord.
40
In the context of circle theorems, which of the following is a correct statement regarding alternate segment theorem?
Answer
(A)
The angle between a tangent and a chord is equal to the angle in the opposite segment.
41
In a circle, which relationships can be used to find angle measures?
Answer
A
B
C
D
42
If a quadrilateral is inscribed in a circle, what must be true about its opposite angles?
Answer
(C)
They are supplementary.
43
If a line segment is drawn from the center of a circle to the midpoint of a chord, what is the relationship between that line segment and the chord?
Answer
(B)
It is perpendicular to the chord.
44
Which of the following statements is true about the angles in the same segment of a circle?
Answer
(C)
They are equal.
45
Given a circle, what is the relationship between an angle formed by two tangents at the external point and the angle formed at the centre by the radius that meets the tangent?
Answer
(C)
They are complementary.
46
Which of the following can be used to determine the length of a chord?
Answer
A
B
C
D
47
In a cyclic quadrilateral, if one of the angles is 70 degrees, what is the measure of the opposite angle?
Answer
(C)
110 degrees
48
Which properties must be true in a cyclic quadrilateral?
Answer
(B)
Opposite angles must add up to 180 degrees.
49
What is the relationship between tangents drawn from an external point to a circle?
Answer
(C)
They are equal in length.
50
In a circle, if two chords are equal in length, what can you conclude about their distance from the center?
