Circle theorems
9 flashcards to master Circle theorems
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The angle at the centre of a circle is 130°. What is the angle at the circumference subtended by the same arc?
The angle at the centre is twice the angle at the circumference when subtended by the same arc. Therefore, the angle at the circumference is 130°/2 = 65°.
State the angle in a semicircle theorem.
The angle in a semicircle is always a right angle (90°). This is a special case of the 'angle at the centre' theorem, where the angle at the centre is 180°.
What is a cyclic quadrilateral, and what is the relationship between its opposite angles?
A cyclic quadrilateral is a quadrilateral whose vertices all lie on the circumference of a circle. The opposite angles of a cyclic quadrilateral add up to 180°.
A tangent to a circle meets the radius at what angle?
A tangent to a circle is perpendicular to the radius at the point of contact. This means they meet at an angle of 90°.
Describe the alternate segment theorem.
The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. The alternate segment is the area of the circle cut off by the chord.
Angles subtended by the same chord in the same segment of a circle are what?
Angles subtended by the same chord, on the same side of it, are equal. Visualize drawing two different triangles from the chord to the circumference; the angles at the circumference will be equal.
A line is drawn from the centre of a circle to bisect a chord. What is the angle between the line and the chord?
A line drawn from the centre of a circle to bisect a chord is perpendicular to the chord. Therefore, the angle is 90°.
In a cyclic quadrilateral ABCD, angle A = 70°. What is the size of angle C?
In a cyclic quadrilateral, opposite angles are supplementary (add up to 180°). Therefore, angle C = 180° - 70° = 110°.
A tangent touches a circle at point T. Chord AB is drawn from T. If the angle between the tangent and chord AB is 60°, what is the angle at the circumference in the alternate segment, opposite angle ATB?
According to the Alternate Segment Theorem, the angle between the tangent and the chord equals the angle in the alternate segment. So, the angle in the alternate segment is also 60°.
Key Questions: Circle theorems
State the angle in a semicircle theorem.
The angle in a semicircle is always a right angle (90°). This is a special case of the 'angle at the centre' theorem, where the angle at the centre is 180°.
What is a cyclic quadrilateral, and what is the relationship between its opposite angles?
A cyclic quadrilateral is a quadrilateral whose vertices all lie on the circumference of a circle. The opposite angles of a cyclic quadrilateral add up to 180°.
A tangent to a circle meets the radius at what angle?
A tangent to a circle is perpendicular to the radius at the point of contact. This means they meet at an angle of 90°.
Describe the alternate segment theorem.
The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. The alternate segment is the area of the circle cut off by the chord.
About Circle theorems (4.7)
These 9 flashcards cover everything you need to know about Circle theorems for your Cambridge IGCSE Mathematics (0580) exam. Each card is designed based on the official syllabus requirements.
What You'll Learn
- 4 Definitions - Key terms and their precise meanings that examiners expect
- 2 Key Concepts - Core ideas and principles from the 0580 syllabus
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Continue Learning
After mastering Circle theorems, explore these related topics:
- 4.6 Circles - 9 flashcards
- 4.8 Similarity and congruence - 9 flashcards
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