One Rule, Then a Whole Vocabulary
A circle is the simplest figure to define and the richest to explore. The whole thing is built from a single sentence: fix a point O, the center, and a length r, the radius; the circle is every point that sits exactly distance r from O. Nothing more. In the language you met earlier, a circle is a locus — the set of all points obeying one condition — and that one condition, distance-to-center equals r, is the seed from which every theorem in this rung will grow.
Because every point is the same distance from O, all the radii of one circle are equal. That sounds too obvious to state, yet it is the single fact most proofs about circles secretly lean on: the moment you draw two radii, you have handed yourself two equal segments, and equal segments breed isosceles triangles, congruences, and symmetry. Keep that reflex — when stuck inside a circle, draw a radius — and half the work is already done.
Before the theorems, fix the names, because circle problems are half won by reading the diagram correctly. A chord is any segment joining two points on the circle. A diameter is the special chord that passes through O — the longest chord there is, and exactly twice the radius. A line that meets the circle at two points is a secant, and one that grazes it at a single point is a tangent. The full naming you set up in the circle's anatomy is the toolkit; here we put each part to work.
Chords and Arcs: A Faithful Partnership
Every chord cuts the circle into two pieces of curve, called arcs — a shorter minor arc and a longer major arc (unless the chord is a diameter, when the two arcs are equal semicircles). The chord is the straight shortcut; the arc is the curved path between the same two endpoints. We measure an arc not in length but in degrees: an arc's degree measure is the central angle that opens onto it from O. A quarter of the circle is a 90-degree arc, a semicircle is 180 degrees, the whole way round is 360.
Now the partnership. Draw two chords of the same circle that happen to be equally long. Connect each chord's endpoints to O, and you get two triangles whose three pairs of sides match: two radii on each (all equal) and the equal chords as the third side. By the side-side-side idea you already trust from the triangle rung, the triangles are congruent, so their central angles are equal — and equal central angles mean equal arcs. In one breath: in a given circle, equal chords subtend equal arcs, and equal arcs are subtended by equal chords. The converse runs just as cleanly.
The Perpendicular from the Center
Here is the single most useful chord fact, and it is the one to drill into your hands. Drop a line from the center O straight down onto a chord so that it meets the chord at a right angle. That perpendicular always lands on the midpoint of the chord — it bisects it. And running the idea backwards is just as true: the line from O to a chord's midpoint is automatically perpendicular to the chord, and the perpendicular bisector of any chord always passes through the center. Three statements, one picture.
Why must it be so? Join O to both endpoints of the chord with radii. The triangle you have built is isosceles — two of its sides are radii, hence equal. The perpendicular from the apex O to the base is the altitude of an isosceles triangle, and you proved back in the triangle rung that this altitude lands dead on the midpoint and bisects the apex angle. So the right-angle line, the midpoint, and the symmetry are not three coincidences; they are one isosceles triangle wearing a circle.
- A chord of length 16 sits in a circle of radius 10. How far is the chord from the center? Drop the perpendicular from O; it bisects the chord into two halves of 8.
- Now look at the right triangle made of: half the chord (8), the radius to an endpoint (10, the hypotenuse), and the unknown distance d from center to chord.
- Apply the Pythagorean theorem: d^2 + 8^2 = 10^2, so d^2 = 100 - 64 = 36, giving d = 6. The chord lies 6 units from the center.
- Read the lesson sideways: the closer a chord is to the center, the longer it is, and the diameter (distance 0) is the longest of all. Distance from center and chord length march in opposite directions.
The Tangent: A Line That Only Just Touches
A tangent is a line that touches the circle at exactly one point — the point of tangency — and nowhere else. Picture a coin resting on a table: the tabletop is tangent to the rim, kissing it at a single spot. The governing law is short and powerful: at the point of tangency, the tangent line is perpendicular to the radius drawn to that point. Radius and tangent meet at a clean 90 degrees, always.
The reason is a tidy little argument by contradiction. Of all the points on the tangent line, the point of tangency is the only one actually on the circle; every other point of the line is outside, hence farther than r from O. So among all points of the line, the point of tangency is the closest to O. But the shortest segment from a point to a line is the perpendicular one — that is exactly the distance from a point to a line. The closest point being the foot of the perpendicular forces the radius to that point to be perpendicular to the line. Done.
Two consequences earn their keep. First, from a point P outside the circle you can draw exactly two tangent lines, and the two tangent segments from P to the points of tangency are equal in length — a fact you can see by joining P to O and noticing two congruent right triangles sharing the hypotenuse PO. Second, since the tangent meets the radius at a right angle, you can chase tangent problems with the Pythagorean theorem on triangle OPT, where T is the point of tangency: |OP|^2 = r^2 + |PT|^2. The tangent quietly becomes a right-triangle problem.
Arcs You Can Measure, and an Honest Word on Length
We have measured arcs in degrees, but an arc is also a real curved length you could walk along, and the two are easy to confuse. A 90-degree arc is one quarter of the way around the circle, so its length is one quarter of the whole circumference 2 pi r. In general, an arc of n degrees has length (n / 360) times 2 pi r — degree measure tells you the fraction, the circumference supplies the size. Keep the two ideas apart: degrees describe what slice of turning the arc spans; length describes how far you would travel.
There is a cleaner unit hiding here that you will lean on for the rest of mathematics. Instead of degrees, measure the central angle in radians — the angle whose arc length equals one radius. Then arc length becomes simply r times theta, with no 360 and no pi cluttering the formula. A full turn is 2 pi radians precisely because the whole circumference is 2 pi r, exactly 2 pi radii laid end to end. Radians are not a different fact, just the honest unit in which the circle's own geometry writes itself.