Mathematics · Simulation and virtual lab Free
Circle theorems: drag the points and measure the angles
Drag points around a circle, watch the angles change and discover the circle theorems for yourself.
- You will learn to
- Discover the circle theorems by measuring angles.
- Use the theorems with correct reasons to find unknown angles.
- Recognise reflex angles at the centre and supplementary angles in cyclic quadrilaterals.
- You will measure
- Angles at the centre and at the circumference (°); tangent lengths (cm).
- Time needed
- About 30–40 min
- Equipment
- Nothing extra. Real version: compasses, protractor, ruler, pencil.
- Price
- Free: no payment, no account.
Virtual lab
Follow the steps with the simulation above (go back to the simulation ↑). Each time you tap Record, a row is added to the table and a point to the graph.
Aim
To discover and verify the circle theorems by measuring angles, and to use them to find unknown angles.
What to do
- Choose Angle at the centre. Drag A, B and P. Record five different positions, including one where ∠AOB is reflex.
- Look at the graph of ∠AOB against ∠APB. What is the gradient?
- Choose Angle in a semicircle and drag P around the circle. Record three positions.
- Choose Angles in the same segment. Keep P and Q on the same side of AB, then move Q to the other side. What changes?
- Choose Cyclic quadrilateral and Tangents. Record what stays the same when you drag the points.
- Use Challenge: find the angle for each theorem until you get three correct answers in a row.
Results
⚠️ Avoid these mistakes
- Using the angle at the centre on the wrong arc: it must stand on the same arc as the angle at the circumference.
- Adding adjacent angles of a cyclic quadrilateral: only OPPOSITE angles add to 180°.
- Giving an answer without a reason: always name the theorem.
✍️ Conclusion
Write your own conclusion first: what do your results and graph show, and which law or principle do they support?
Check your conclusion
The angle at the centre is twice the angle at the circumference on the same arc, so the angle in a semicircle is 90°. Angles in the same segment are equal, opposite angles of a cyclic quadrilateral add up to 180°, a tangent is perpendicular to the radius, tangents from an external point are equal, and the tangent–chord angle equals the angle in the alternate segment.
Questions
State the angle at the centre theorem.
The angle subtended by an arc at the centre of a circle is twice the angle it subtends at any point on the remaining part of the circumference.
What is the angle in a semicircle?
90°: the angle subtended by a diameter at any point on the circumference is a right angle.
What do you know about angles in the same segment?
They are equal: angles subtended by the same chord on the same side of it are equal.
Two opposite angles of a cyclic quadrilateral are x and 3x. Find x.
x + 3x = 180°, so x = 45° and the other angle is 135°.
What is the angle between a tangent and the radius at the point of contact?
90°: the tangent is perpendicular to the radius.
State the alternate segment theorem.
The angle between a tangent and a chord at the point of contact equals the angle subtended by the chord in the alternate segment.
For teachers: expected results and how the model works
Points move on a circle of radius 5 cm (118 design units); angles are measured exactly and shown to the nearest degree, so ratios may differ slightly from 2.00 because of rounding. When P is on the minor arc, ∠AOB is shown as a reflex angle. In “same segment”, moving Q across AB shows the supplementary case (cyclic quadrilateral). Challenge mode hides one angle; answers within 1° are accepted.
Quick check 3 questions · scored
Choose one answer for each question. You see at once if it is right, why, and your score at the end.
1. ∠AOB = 140° at the centre. What is ∠APB on the major arc?
The angle at the circumference is half the angle at the centre: 70°.
2. One angle of a cyclic quadrilateral is 105°. What is the opposite angle?
Opposite angles add up to 180°: 180 − 105 = 75°.
3. OT = 5 cm is a radius and PT is a tangent; OP = 13 cm. PT = ?
Tangent ⟂ radius, so PT² = 13² − 5² = 144 and PT = 12 cm.
Your score:
Exam practice Exam set
National-exam style questions on this simulation (11 marks). Try each one on paper first, then open the marking scheme.
O is the centre of a circle. A, B and P are on the circumference and ∠AOB = 124°. Find ∠APB, giving a reason. (2 marks)
Marking scheme: ∠APB = 124° ÷ 2 = 62° (1); the angle at the centre is twice the angle at the circumference (1).
ABCD is a cyclic quadrilateral with ∠DAB = 3x + 10° and ∠BCD = 2x − 5°. Find x and ∠DAB. (3 marks)
Marking scheme: 3x + 10 + 2x − 5 = 180 (1); 5x = 175, x = 35 (1); ∠DAB = 115° (1).
From a point P outside a circle of centre O and radius 5 cm, a tangent PT is drawn with PO = 13 cm. Calculate PT. (3 marks)
Marking scheme: ∠OTP = 90° (tangent ⟂ radius) (1); PT² = 13² − 5² (1); PT = 12 cm (1).
AB is a diameter of a circle and C is a point on the circumference with ∠CAB = 35°. Find ∠ABC, giving reasons. (3 marks)
Marking scheme: ∠ACB = 90° (angle in a semicircle) (1); ∠ABC = 180° − 90° − 35° (1) = 55° (angles in a triangle) (1).
Donat Sciences and MathsMathematics · Virtual lab worksheet
School: ........................................ Name: ................................ Class: ........ Date: ............
Circle theorems: drag the points and measure the angles
Aim: To discover and verify the circle theorems by measuring angles, and to use them to find unknown angles.
Apparatus (real laboratory): compasses, protractor, ruler, pencil.
What to do (with the simulation on www.donatstem.co.rw)
- Choose Angle at the centre. Drag A, B and P. Record five different positions, including one where ∠AOB is reflex.
- Look at the graph of ∠AOB against ∠APB. What is the gradient?
- Choose Angle in a semicircle and drag P around the circle. Record three positions.
- Choose Angles in the same segment. Keep P and Q on the same side of AB, then move Q to the other side. What changes?
- Choose Cyclic quadrilateral and Tangents. Record what stays the same when you drag the points.
- Use Challenge: find the angle for each theorem until you get three correct answers in a row.
Results
| # | Theorem | First angle (°) | Second angle (°) | Relation |
|---|
| 1 | | | | |
| 2 | | | | |
| 3 | | | | |
| 4 | | | | |
| 5 | | | | |
| 6 | | | | |
| 7 | | | | |
| 8 | | | | |
| 9 | | | | |
| 10 | | | | |
| 11 | | | | |
| 12 | | | | |
Graph
Plot ∠AOB (vertical axis) against ∠APB for your “angle at the centre” results and find the gradient.
Questions
- State the angle at the centre theorem. [2]
- What is the angle in a semicircle? [2]
- What do you know about angles in the same segment? [2]
- Two opposite angles of a cyclic quadrilateral are x and 3x. Find x. [2]
- What is the angle between a tangent and the radius at the point of contact? [2]
- State the alternate segment theorem. [2]
- O is the centre of a circle. A, B and P are on the circumference and ∠AOB = 124°. Find ∠APB, giving a reason. [2]
- ABCD is a cyclic quadrilateral with ∠DAB = 3x + 10° and ∠BCD = 2x − 5°. Find x and ∠DAB. [3]
- From a point P outside a circle of centre O and radius 5 cm, a tangent PT is drawn with PO = 13 cm. Calculate PT. [3]
- AB is a diameter of a circle and C is a point on the circumference with ∠CAB = 35°. Find ∠ABC, giving reasons. [3]
Donat Sciences and MathsMathematics · Virtual lab worksheet
School: ........................................ Name: ................................ Class: ........ Date: ............
Marking scheme: Circle theorems: drag the points and measure the angles
Expected results
Points move on a circle of radius 5 cm (118 design units); angles are measured exactly and shown to the nearest degree, so ratios may differ slightly from 2.00 because of rounding. When P is on the minor arc, ∠AOB is shown as a reflex angle. In “same segment”, moving Q across AB shows the supplementary case (cyclic quadrilateral). Challenge mode hides one angle; answers within 1° are accepted.
Answers
- State the angle at the centre theorem. [2]
The angle subtended by an arc at the centre of a circle is twice the angle it subtends at any point on the remaining part of the circumference.
- What is the angle in a semicircle? [2]
90°: the angle subtended by a diameter at any point on the circumference is a right angle.
- What do you know about angles in the same segment? [2]
They are equal: angles subtended by the same chord on the same side of it are equal.
- Two opposite angles of a cyclic quadrilateral are x and 3x. Find x. [2]
x + 3x = 180°, so x = 45° and the other angle is 135°.
- What is the angle between a tangent and the radius at the point of contact? [2]
90°: the tangent is perpendicular to the radius.
- State the alternate segment theorem. [2]
The angle between a tangent and a chord at the point of contact equals the angle subtended by the chord in the alternate segment.
- O is the centre of a circle. A, B and P are on the circumference and ∠AOB = 124°. Find ∠APB, giving a reason. [2]
∠APB = 124° ÷ 2 = 62° (1); the angle at the centre is twice the angle at the circumference (1).
- ABCD is a cyclic quadrilateral with ∠DAB = 3x + 10° and ∠BCD = 2x − 5°. Find x and ∠DAB. [3]
3x + 10 + 2x − 5 = 180 (1); 5x = 175, x = 35 (1); ∠DAB = 115° (1).
- From a point P outside a circle of centre O and radius 5 cm, a tangent PT is drawn with PO = 13 cm. Calculate PT. [3]
∠OTP = 90° (tangent ⟂ radius) (1); PT² = 13² − 5² (1); PT = 12 cm (1).
- AB is a diameter of a circle and C is a point on the circumference with ∠CAB = 35°. Find ∠ABC, giving reasons. [3]
∠ACB = 90° (angle in a semicircle) (1); ∠ABC = 180° − 90° − 35° (1) = 55° (angles in a triangle) (1).