Mathematics for IGCSE & O level - Circle Theorems (Section 2)
1
What is the property of angle subtended at the center by an arc?
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(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?
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(B)
They are supplementary.
3
What is the relationship between the lengths of the tangents drawn from an external point to a circle?
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(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?
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(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?
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(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?
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(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?
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A
C
D
8
What is the relationship between the lengths of the tangents from an external point to a circle?
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(A)
Equal
9
What type of angle is formed in a semicircle?
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(C)
A right angle
10
Which of the following statements about angles in circles are correct?
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A
C
11
What angle is formed by a tangent and a radius at the point of tangency?
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(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?
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(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?
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(C)
90 degrees
14
Which of the following is a key characteristic of a cyclic quadrilateral?
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(C)
Opposite angles are supplementary.
15
Which of the following is the converse of the theorem stating 'Equal chords are equidistant from the center'?
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(A)
Chords equidistant from the center are equal.
16
What relationship exists between the angles marked with letters in the circles?
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A
B
C
17
What type of angle is inscribed in a semicircle?
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(C)
Right
18
If two tangents are drawn from a point outside a circle, what is always true about their lengths?
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(C)
They are equal in length.
19
Which property relates to the lengths of tangents drawn from a point outside the circle?
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(A)
They are equal.
20
If two circles touch externally, how many common tangents can be drawn?
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(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?
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(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?
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(C)
90°
23
Two tangents are drawn from an external point to a circle. What is true about the lengths of these tangents?
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(B)
They are equal in length.
24
Which statement correctly describes the property of angles in the same segment of a circle?
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(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?
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(A)
35 degrees
26
In a circle, if two chords are the same length, how are they positioned relative to the center?
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(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?
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(C)
They are equal.
28
What is the relationship between the angles in the same segment of a circle?
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(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?
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(C)
It's always a kite.
30
Which of the following must be true in a circle?
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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?
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(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?
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(B)
They are perpendicular.
33
What can you deduce about the chords which are equidistant from the centre of a circle?
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(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?
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(B)
They are perpendicular.
35
What can be concluded if two circles touch externally?
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(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?
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(B)
They are congruent.
37
What is the necessary and sufficient condition for a quadrilateral to be cyclic?
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(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?
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(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?
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(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?
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(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?
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A
B
C
D
42
If a quadrilateral is inscribed in a circle, what must be true about its opposite angles?
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(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?
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(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?
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(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?
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(C)
They are complementary.
46
Which of the following can be used to determine the length of a chord?
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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?
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(C)
110 degrees
48
Which properties must be true in a cyclic quadrilateral?
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(B)
Opposite angles must add up to 180 degrees.
49
What is the relationship between tangents drawn from an external point to a circle?
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(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?
