Home › Worksheets › IGCSE Year 10 Math › Circle Geometry Prepared by Mohsim Digital Education
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The angle at the centre is double the angle at the circumference on the same arc, and any angle in a semicircle sits at 90°, so circle theorems run straight through this Year 10 sheet. The key also fixes the tangent as perpendicular to the radius at contact.
Key Takeaways
The angle at the centre is double the angle at the circumference on the same arc (Q1).
The angle in a semicircle is 90° (Q2).
The tangent at a point is perpendicular to the radius at the point of contact.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.
The angle at the centre of a circle is what compared to the angle at the circumference subtended by the same arc?
(a) half (b) equal to (c) double (d) three times
2.
What is the size of the angle in a semicircle?
(a) 60° (b) 90° (c) 45° (d) 180°
3.
Angles in the same segment of a circle are what?
(a) equal (b) supplementary (c) complementary (d) unrelated
4.
The tangent to a circle is perpendicular to which line?
(a) the chord (b) the radius at the point of contact (c) any diameter (d) the circumference
5.
Opposite angles of a cyclic quadrilateral sum to what?
(a) 90° (b) 180° (c) 360° (d) 120°
6.
The angle at the centre subtending an arc is 140°. What is the angle at the circumference subtending the same arc?
(a) 70° (b) 140° (c) 35° (d) 280°
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.
State the angle at the centre theorem in words.
8.
A quadrilateral is cyclic. Two of its opposite angles are 80° and 100°. State what is special about opposite angles of a cyclic quadrilateral.
9.
State the angle between a tangent and the radius at the point of contact.
10.
What is the size of an angle in a semicircle?
11.
Two angles in the same segment have sizes 47° and x°. Find x.
12.
A tangent of length 12 cm is drawn from a point P to a circle of radius 5 cm. Find the distance from P to the centre.
13.
The angle at the centre subtending an arc is 100°. Find the angle at the circumference subtending the same arc.
14.
A chord of length 8 cm is 3 cm from the centre of a circle. Find the radius.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.
A tangent from an external point P touches a circle at T. The length PT is 8 cm and the radius OT is 6 cm. Find the length OP.
16.
A and B are points on a circle and the angle at the centre AOB is 120°. Find the angle ACB where C is a point on the major arc AB.
17.
A cyclic quadrilateral ABCD has angle A = 110°. Find the size of the opposite angle C.
18.
Two tangents are drawn to a circle from an external point. Find the angle between each tangent and the radius at its point of contact.
19.
A chord of length 24 cm is 5 cm from the centre of a circle. Find the radius.
20.
A, B and C lie on a circle with AB as a diameter and angle BAC = 40°. Find the size of angle ABC.
Answer Key
1. c) double
2. b) 90°
3. a) equal
4. b) the radius at the point of contact
5. b) 180°
6. a) 70°
7. Refer to solution guide.
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End of Worksheet
People Also Ask
How does the angle at the centre compare with the angle at the circumference?
It is double, option (c) in this worksheet (Q1).
What is the size of the angle in a semicircle?
90°, option (b) in this worksheet (Q2).
What is the tangent at a point perpendicular to?
The radius at the point of contact.