Circle Theorem Circus

Circle Theorem Circus

Welcome to the Higher-tier big top. Circle theorems look intimidating, but there are only a handful, and each one is a single, quotable fact. The winning routine every time: spot which theorem the diagram is showing, apply it to get the angle, and name it. A circle-theorem answer with no reason drops marks, and this module tests that habit rather than only mentioning it.

Know the parts of a circle

The theorems are built out of these words, so lock them in first. Note that a diameter is a special chord, not a separate kind of line:

Tap the diameter

Four lines are drawn on this circle. Tap the DIAMETER: the straight line that passes through the centre.

Name the parts

A straight line joining two points on the circle is a _____, and if it happens to pass through the centre it is also a _____. A line that touches the circle at exactly one point is a _____. The region between two radii and an arc is a _____.

Centre, and semicircle

The first theorem, and the special case that comes out of it. The second is the one you will use most, because a diameter in a diagram is very easy to spot:

Angle at the centre

Two points on a circle subtend an angle of 40° at the circumference. How many degrees is the angle they subtend at the centre?

Why must it be 90°?

The angle in a semicircle is always a right angle. Which argument shows that this follows from the angle-at-the-centre theorem, rather than being an unrelated rule to memorise?

• A diameter is a straight line through the centre, so it subtends 180° there, and the angle at the circumference must be half of that
• The angles of a triangle add to 180°, and the other two are equal, so each is 45° and the third is 90°
• It has been measured in many triangles and always comes out at 90°
• The circle is symmetrical, so any angle drawn inside it must be a right angle

The rest of the act

Four more, and each is one quotable sentence: • Angles in the same segment are equal: two angles at the circumference standing on the same chord are the same size. • A cyclic quadrilateral has all four corners on the circle, and its opposite angles add to 180°. • A tangent meets a radius at 90° at the point of contact. • Two tangents from the same external point are equal in length, so the figure they make with the two radii is a kite. Higher-tier questions rarely use one of these alone. They chain two or three, and every link in the chain needs naming.

Cyclic quadrilateral

A quadrilateral has all four vertices on a circle. One angle is 85°. How many degrees is the angle OPPOSITE to it?

Which theorem is this diagram asking for?

A two-step chase

This is what a Higher-tier question actually looks like, and it is worth reading slowly. *AB is a diameter of a circle with centre O. C is a point on the circumference, and angle CAB = 28°. Find angle ABC.* Step 1. Angle ACB = 90°, because the angle in a semicircle is a right angle. Step 2. The angles of triangle ABC add to 180°, so angle ABC = 180° − 90° − 28° = 62°. Two facts, both named. Notice that the naming is not decoration: those are the method marks, and a bare "62°" collects fewer of them than the same number with its two reasons attached.

Run the chase

PQ is a diameter of a circle. R is a point on the circumference, and angle RPQ = 34°. How many degrees is angle PQR?

Name your reason

In the chase you just did, you used the fact that angle PRQ = 90°. Which reason must be quoted alongside it to earn the method mark?

• The angle in a semicircle is 90°
• A tangent meets a radius at 90°
• Angles in the same segment are equal
• Angles in a triangle add to 180°

Which are true?

Select ALL THREE statements that are TRUE.

• The angle in a semicircle is a special case of the angle-at-the-centre theorem, because a diameter subtends a straight angle of 180° at the centre
• Two tangents from the same external point are equal in length, so the figure formed with the two radii is a kite
• A circle-theorem answer can lose marks even when the number is correct, if no theorem is named
• The angle at the centre is half the angle at the circumference
• Every chord of a circle passes through the centre
• Opposite angles of any quadrilateral add to 180°

Quote the theorems

The angle at the centre is _____ the angle at the circumference standing on the same arc. The angle in a semicircle is _____ degrees. Opposite angles of a cyclic quadrilateral add to _____ degrees. A tangent meets a _____ at a right angle where it touches the circle.