🎯Angles & Segments in Circles

Keeper Orin measures the round tarn at Circle Basin with nothing but a cord, and every question in this skill is really about six named parts of a circle. Learn them once and the theorems stop sounding like a list.

• ◆A radius runs from the centre to the rim. Every radius of one circle is the same length, and that single fact hides inside almost every proof here.
• ◆A chord joins two points on the rim without going through the centre.
• ◆A diameter is the longest chord: it passes through the centre and is two radii long.
• ◆An arc is a piece of the rim. Its size is measured in degrees, not in centimetres, and the whole rim is 360°.
• ◆A tangent is a line that touches the circle at exactly one point and never enters it.
• ◆A secant is a line that cuts straight through, crossing the rim twice.

Two arcs join any pair of points A and B, so an arc always has to be named carefully. The short one is the minor arc, the long one is the major arc, and when a question says "arc AC, the arc that does not pass through B", it is telling you exactly which of the two it means.

Points are named with capital letters and O is always the centre. "Angle ABC" means the angle whose corner is at the middle letter, B, with arms running out to A and to C. Every question in this skill names its points that way, so read the middle letter first: it tells you where you are standing.

An angle standing on the arc AC can have its corner in two useful places.

• ◆Central angle. The corner is at the centre O, so the arms are two radii. A central angle is simply *equal* to its arc: if angle AOC is 84°, then arc AC is 84°. Central angle and arc are two names for one number.
• ◆Inscribed angle. The corner is on the rim, at some third point B. An inscribed angle is half of the arc it stands on. If arc AC is 84°, then angle ABC is 42°.

That is the whole inscribed angle theorem, and it comes with three consequences worth memorising:

• ◆Same arc, same angle. Move B anywhere along the far side of the circle and angle ABC does not change, because the arc it stands on has not changed. Two inscribed angles on the same arc are equal.
• ◆The semicircle. If AB is a diameter, the arc it stands on is exactly half of 360°, so any angle ACB drawn from the rim is 180 / 2 = 90°. A diameter always produces a right angle. Once you have that right angle you have a right triangle, and the other two angles must add to 90°.
• ◆The cyclic quadrilateral. If all four vertices of ABCD lie on one circle, angle ABC and angle CDA stand on the two arcs from A to C, which together make 360°. Half of 360° is 180°, so opposite angles of an inscribed quadrilateral add to 180°.

The trap in every one of these is direction. Going from the centre outwards to the rim you halve; going from the rim back in to the centre you double. Ask yourself which end you started at before you touch the number.

One more piece of care: an inscribed angle stands on the arc it does *not* touch. If B sits on the minor arc AC, then angle ABC stands on the major arc, which you get by taking the minor arc away from 360° first.

Two facts turn tangent and chord questions into ordinary Pythagorean theorem questions.

A tangent meets the radius at the point of contact at 90°. So if PT touches the circle at T and O is the centre, triangle OTP has a right angle at T, with OP — the segment from the centre out to the external point — as its hypotenuse:

OP² = OT² + PT²

That means you can find the tangent length from the radius and OP, or the distance OP from the radius and the tangent. It also fixes the angles: the three angles of triangle OTP add to 180°, and one of them is already used up, so the other two add to 90°.

Two tangents drawn from the same outside point are equal. PA and PB from the point P are the matching sides of two identical right triangles that share the hypotenuse OP, so PA = PB. The same picture gives a four-sided figure OAPB whose angles add to 360°, two of which are the right angles at A and B — so angle AOB and angle APB always add to 180°.

A perpendicular from the centre bisects a chord. Drop a line from O onto the chord AB at right angles and it lands at M, the exact midpoint. That splits the chord into two equal halves and builds a right triangle with legs (half the chord) and (the distance OM) and the radius as hypotenuse:

radius^2 = OM² + (AB / 2)²

The half is what students forget. If a question gives you the whole chord, halve it before it goes into the formula; if a question asks for the whole chord, double at the end. And because that triangle fixes the distance completely, equal chords are the same distance from the centre, and chords the same distance from the centre are equal.

Two straight lines cutting a circle make an angle, and that angle is decided entirely by the arcs they cut off. There are only two cases, and the difference between them is a sign.

Crossing inside the circle — two chords meeting at a point P inside. The angle is half the sum of the two arcs it cuts off, the one it faces and the one behind it:

angle P = (arc 1 + arc 2) / 2

Meeting outside the circle — two secants, a secant and a tangent, or two tangents, all from a point P outside. The angle is half the difference of the far arc and the near arc:

angle P = (far arc - near arc) / 2

A quick way to keep them straight: as P slides outwards the angle shrinks, and only the difference version can shrink to nothing. Inside means add; outside means subtract.

The same two positions also control lengths, through three product rules:

• ◆Two chords crossing inside: PA times PB = PC times PD, where PA and PB are the two pieces of one chord and PC and PD the two pieces of the other.
• ◆Two secants from outside: PA times PB = PC times PD, where PA is the external part of the first secant and PB is the whole secant measured all the way from P — external part times whole secant, for each line.
• ◆A tangent and a secant from outside: PT² = PA times PB, with PA the external part and PB the whole secant. The tangent is squared because a tangent is a secant whose two crossing points have slid together, so it plays both parts at once.

If a question gives you the chord AB inside the circle rather than the whole secant PB, add the external part on first: PB = PA + AB. Using the chord where the whole secant belongs is the most common wrong answer in this whole skill.

Every prompt in this skill is complete in words — the points are all named, and every arc is described by its endpoints and by which point it does *not* pass through. Where you see a picture, it is a circle with one shaded wedge, and that wedge is the central angle the prompt gives you. Everything else has to be built in your head or on scrap paper, which is exactly what a Geometry exam asks for too.

Angles and arcs are whole numbers of degrees. Type just the number, 47, rather than "47 degrees" — the degree sign is accepted too, but it changes nothing.

Lengths are in centimetres. At the easier difficulties every length is a whole number, because the right triangles are built from Pythagorean triples such as 3-4-5, 5-12-13 and 8-15-17. When a length does not come out whole, the prompt says "round to 1 decimal place" — answer 15.9, and anything within a tenth is accepted, so it does not matter whether you rounded at the end or carried extra digits.

Algebraic questions give two angles as expressions such as (2x + 15)° and (3x)°. Write the equation the theorem gives you — sum 180° for opposite angles of a cyclic quadrilateral, sum 90° for the two acute angles beside a semicircle's right angle — solve for x, and then put x back in, because the question asks for the angle, not for x.

Reasoning questions give four statements and ask which one *must* be true. Choose the option, not a number. "Must be true" means true for every circle you could draw, so a statement that happens to hold in one neat picture but not in a stretched one is the wrong answer.