Chapter 12.2 — Exercise 12.1 — Circle Basics

Reading Eight Parts Off One Figure

Circle with centre O has points A, B, C, D marked on and around it, with AB a straight line through O.

Every single answer in this table traces back to just three definitions from the introduction: a segment joining two circle points is a chord unless it passes through the centre, in which case it's specifically a diameter; an arc is minor or major depending purely on which of the two pieces is shorter; and a segment is whichever region a chord and its matching arc enclose together. Nothing here requires measuring anything at all — each identification follows purely from reading the figure carefully against a fixed definition, the same skill tested throughout the rest of this exercise. Two of the eight entries above are worth a second look because they're easy to mix up on a first pass: AC and AD are both labeled chords even though they look like they should behave differently, one crossing near the diameter and one further from it — but "chord" never distinguishes based on how close a segment happens to sit to the diameter or the centre, only on whether it joins two points on the circle without itself passing through O. A segment can be extremely short, extremely long, or anywhere in between and still qualify as a chord the moment both its endpoints land on the circle's boundary.

Seven Statements, Testing the Definitions Directly

The two "False" statements are false for genuinely different reasons, worth telling apart rather than just memorizing as wrong. "A diameter divides the circle into two unequal parts" contradicts a proved fact directly — a diameter always produces two equal semicircles, precisely because it passes through the centre, so "unequal" is simply the wrong word choice, not a subtler misunderstanding. "A sector is the area enclosed by two radii and a chord," though, contains a genuinely common mix-up: a sector is enclosed by two radii and an arc, not a chord — swapping "arc" for "chord" quietly turns the definition of a sector into the definition of a triangle-plus-segment instead, a completely different region. It's worth noticing that this particular false statement is really testing the same "arc, not chord" distinction that separates a sector from a segment in the first place, first raised back in the introduction — a sector's two straight boundaries are always radii meeting at the centre, never a single chord cutting straight across, and a segment's boundary is exactly the reverse: one chord plus one arc, with no radius involved anywhere in its definition at all.

The five "True" statements split into two groups worth noticing separately: three of them (the plane-division statement, and the two segment definitions) are pure vocabulary, restating a definition from the introduction with no independent fact folded in. The other two — "longest chord is the diameter" and "midpoint of a diameter is the centre" — are genuinely provable facts rather than restated definitions: the diameter's length as the maximum possible chord follows from every other chord failing to reach the full width of the circle, and a diameter's midpoint being the centre follows directly from the centre being, by definition, equidistant from both of the diameter's endpoints. Worth stating plainly, since it's easy to blur the two together when reading quickly: a definition can always be checked purely by re-reading the original wording, while a provable fact needs an actual argument connecting it back to definitions already established — recognising which kind of statement is in front of you is itself a small but genuinely useful skill, distinct from knowing the vocabulary itself.

From Naming Parts to Proving Relationships

Every question in this exercise asked for identification, never for a reason beyond a definition. Exercise 12.2 moves to genuine theorems for the first time — connecting a chord's length to the angle it subtends at the centre, the first fact in this chapter that actually needs proving rather than simply naming. Every subsequent exercise keeps circling back to this same set of eight identifications in one form or another — a "find the radius" problem is really and simply asking to recover a length already named right here, and a proof about equal chords is really just a statement about two of the segments defined in this very exercise turning out to be equal to each other. Getting genuinely comfortable naming parts on sight, without hesitating over whether a given segment is a chord or specifically a diameter, is what lets the rest of this chapter's proofs move quickly through their setup and spend their real effort entirely on the actual reasoning instead.

Why the Figure Matters as Much as the List

Reading any circle problem correctly always starts with locating the centre first, since nearly every other identification depends on it: a segment is only a diameter, rather than an ordinary chord, because it happens to pass through that one specific point, and a region only counts as a sector, rather than a segment, because two of its three separate boundary lines happen to meet together at exactly that one point. Skipping past the centre entirely and jumping straight ahead to labelling arcs or chords is a common source of mistakes early on — every definition in this whole chapter is ultimately anchored to that one fixed point, and losing track of exactly where it sits in a busier, more complicated figure is usually where a wrong identification quietly starts.