Circle Theorems flashcards
10 cards on Circle Theorems: the key points, worked examples and the slips to watch. Swipe through them, or print them.
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Circle Theorems
The angle at the centre
How is the angle at the centre linked to the angle at the circumference standing on the same arc?
Answer it in your head, then turn the card over.
The angle at the centre
Key points
- The angle at the centre is twice the angle at the circumference.
- Halve to go out to the circle; double to come in to the centre.
- Both angles must stand on the same arc: their arms meet the circle at the same two points.
- Together they make an arrowhead shape.
Example
Angle at the circumference 48°: at the centre, 2 × 48 = 96°
Reflex angle at the centre 250°: at the circumference, 250 ÷ 2 = 125°
Watch out
Doubling instead of halving: 140° at the centre gives 70° at the circumference, never 280°.
Circle Theorems
Angles in the same segment
What is true about two angles at the circumference that stand on the same arc?
Answer it in your head, then turn the card over.
Angles in the same segment
Key points
- They are equal.
- Each one is half of the same angle at the centre, so they must match.
- Look for a bow tie: two angles drawn from the ends of one chord to points on the circle.
- Quote it as: angles in the same segment are equal.
Watch out
Angles on opposite sides of the chord are not equal: they make a cyclic quadrilateral and add to 180°.
Circle Theorems
The angle in a semicircle
What is the angle in a semicircle, and why?
Answer it in your head, then turn the card over.
The angle in a semicircle
Key points
- A triangle drawn from the ends of a diameter to a point on the circle has a right angle at that point.
- The diameter is a straight line through the centre, so the angle at the centre is 180°, and half of that is 90°.
- A diameter in a diagram is a signal to look for a right angle.
AB is a diameter and angle CAB is 28°
Angle ACB = 90°: the angle in a semicircle is 90°
Angle ABC = 180° − 90° − 28° = 62°: angles in a triangle
Watch out
Only the 90° is fixed: the other two angles share 90°, but they need not be 45° each.
Circle Theorems
Cyclic quadrilaterals
What do you know about the angles of a cyclic quadrilateral?
Answer it in your head, then turn the card over.
Cyclic quadrilaterals
Key points
- A cyclic quadrilateral has all four corners on the circumference of the circle.
- Each pair of opposite angles adds to 180°.
- Angles next to each other have no fixed total; all four still add to 360°.
ABCD is cyclic: A = 102° and B = 74°
C is opposite A: 180° − 102° = 78°
D is opposite B: 180° − 74° = 106°
Check: 102 + 74 + 78 + 106 = 360
Watch out
A quadrilateral with a corner at the centre is not cyclic: check all four corners are on the circle.
Circle Theorems
A tangent meets a radius at 90°
What angle does a tangent make with the radius, and how do you use it?
Answer it in your head, then turn the card over.
A tangent meets a radius at 90°
Key points
- A tangent is a straight line that touches the circle at exactly one point.
- The tangent and the radius to that point meet at 90°.
- Draw the radius in: it often gives a right-angled triangle for Pythagoras or trigonometry.
Radius 5 cm, P is 13 cm from centre O
The tangent from P touches at T, and angle OTP = 90°
PT² = 13² − 5² = 144
PT = = 12 cm
Watch out
Diagrams rarely mark this right angle: draw in the radius and label the 90° yourself.
Circle Theorems
Two tangents from one point
What is true about the two tangents drawn to a circle from the same point outside it?
Answer it in your head, then turn the card over.
Two tangents from one point
Key points
- They are equal in length, from the point to where each touches the circle.
- So the triangle they make with the chord between the touching points is isosceles.
- With the centre O, they make a kite with a right angle at each touching point.
Tangents PA and PB, angle APB = 50°
PA = PB, so angle PAB = angle PBA = (180° − 50°) ÷ 2 = 65°
Kite OAPB: angle AOB = 360° − 90° − 90° − 50° = 130°
Watch out
The equal lengths run from P to the touching points, not from P to the centre.
Circle Theorems
The alternate segment theorem
What does the alternate segment theorem say, and how do you spot it?
Answer it in your head, then turn the card over.
The alternate segment theorem
Key points
- The angle between a tangent and a chord equals the angle in the alternate segment.
- That angle is at the circumference, on the other side of the chord from the tangent angle.
- Spot it: a tangent and a chord meeting on the circle, and a triangle with a corner there.
Tangent at A makes 58° with chord AB
C is on the circle across the chord: angle ACB = 58° (alternate segment theorem)
If angle BAC = 70°, then angle ABC = 180° − 58° − 70° = 52°
Watch out
The equal angle is across the chord, not next to the tangent: trace the chord to find the right triangle.
Circle Theorems
Giving reasons that earn marks
How do you write the reasons in a circle-theorem question so that they earn the marks?
Answer it in your head, then turn the card over.
Giving reasons that earn marks
Key points
- Quote the theorem in full: the mark is for naming the exact fact you used.
- One step, one angle, one reason: write each new angle on the diagram as you find it.
- Ordinary angle facts count as reasons too: angles in a triangle, on a straight line, in an isosceles triangle.
Reasons written in full
Angle ACB = angle ADB: angles in the same segment are equal.
Angle OAP is a right angle: a tangent meets a radius at right angles.
OA = OB: both are radii, so triangle OAB is isosceles.
Watch out
"Circle theorems" on its own is not a reason: it does not say which fact you used.
Circle Theorems
The theorems, in the words to quote
What are the circle theorems, in the words you write as reasons?
Answer it in your head, then turn the card over.
The theorems, in the words to quote
Key points
- The angle at the centre is twice the angle at the circumference.
- Angles in the same segment are equal, and the angle in a semicircle is 90°.
- In a cyclic quadrilateral, opposite angles add up to 180°.
- A tangent meets a radius at 90°, and tangents from a point outside the circle are equal in length.
- The angle between a tangent and a chord equals the angle in the alternate segment.
Circle Theorems
Radii make isosceles triangles
Why do radii matter when you chase angles in a circle?
Answer it in your head, then turn the card over.
Radii make isosceles triangles
Key points
- Every radius of a circle is the same length.
- So a triangle with two radii as sides is isosceles, and its base angles are equal.
- Mark the equal base angles as soon as you see two radii meeting at the centre.
Angle AOB = 124°, C on the major arc
Angle ACB = 124° ÷ 2 = 62°: the angle at the centre is twice the angle at the circumference
Angle OAB = (180° − 124°) ÷ 2 = 28°: OA = OB are radii, so triangle OAB is isosceles
Watch out
Missing the isosceles triangle: two radii from the centre always give equal base angles.
Print these cards
Four cards to an A4 sheet, each the size of a standard A6 revision card.
One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
Put it into practice
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