Where the lines cross tells you which rule to use.
Find x
Two chords cross inside a circle. One is cut into pieces of 3 and 8; the other into pieces of 4 and x. What is x?
When two chords intersect, the products of the two pieces are equal: 3 × 8 = 4 × x, so 24 = 4x and x = 6. D adds instead of multiplying — 3 + 8 = 11 and 11 − 4 = 7 — which is the commonest wrong route. The rule is about products, never sums.
1
First decide where the lines meet.
Inside the circle, on it, or outside — each has its own pair of rules.
The names first: a chord joins two points of the circle, and a chord through the centre is a diameter, the longest chord. A secant is a line that cuts the circle at two points; a tangent touches it at exactly one point.
They meet
Length rule
Angle rule
inside
piece × piece = piece × piece
half the sum of the two arcs
on the circle
—
half the arc (inscribed or tangent-chord)
outside
whole × external = whole × external
half the difference of the two arcs
Inside gives a sum, outside gives a difference. That single contrast decides most questions before any arithmetic starts.
2
Two chords inside: the products match.
Each chord is split by the crossing point into two pieces. Multiply the two pieces
of one chord, do the same for the other, and set them equal.
3
From a point outside: whole × external.
The "whole" runs all the way through the circle, external stop is the near side.
secant 1: external 4, far piece 5
whole = 4 + 5 = 9
secant 2: external 3, far piece y
whole = 3 + y
9 × 4 = (3 + y) × 3
whole × external, both sides
36 = 9 + 3y, so y = 9
the far piece
Whole, not far piece
The first factor is the entire secant from the outside point to
the far side of the circle — external plus far piece. Using only the far piece is
the mistake this rule is built to catch.
4
A tangent is its own whole and its own external piece.
tangent² = whole secant × external piece
A tangent touches at one point only, so it has nothing beyond — its whole and its
external piece are the same length, and "whole × external" becomes the square.
Two secants from outsidearcs 110° and 40° → 110 − 402angle = 35°
Tangent and chordarc 130° → 1302angle = 65°
The outside angle is always the smaller of the pair, because a difference is
smaller than a sum. If your external angle comes out bigger than either arc, the
rule went the wrong way round.
Remember the flow
Where do they meet? · Lengths: products, never sums · Angles: inside adds, outside subtracts · Then halve
A tangent squared is the giveaway that the point is outside the circle.
6
Three right-angle facts about radii, tangents and chords.
radius ⊥ tangent at the point of contacttwo tangents from one point are equalthe perpendicular from the centre bisects a chord
A circle has radius 8 and centre O. A point P is 17 from O. How long is a tangent from P, and the other one?
OT ⊥ PT, so PT² + 8² = 17²
right angle at the point of contact T
PT² = 289 − 64 = 225, PT = 15
8, 15, 17 is a right triangle ✓
second tangent PS = 15
tangents from one point are equal
A circle has radius 13 and a chord of length 24. How far is the chord from the centre?
half the chord = 12
the perpendicular bisects it
d² + 12² = 13²
the radius is the hypotenuse
d = √169 − 144 = 5
check: 5² + 12² = 169 ✓
Halve the chord first
Using the whole chord gives d² = 169 − 576, a negative number — no distance at all. The right triangle has half the chord as one leg, because the perpendicular from the centre cuts the chord into two equal halves.
7
Variations: pieces as expressions, an arc from an angle, two tangents.
The same products and halves, used the other way round.
Pieces as expressions: one chord is cut into x and x + 2, the other into 4 and 6.
x(x + 2) = 4 × 6 = 24
products of the pieces
x² + 2x − 24 = 0, (x + 6)(x − 4) = 0
a quadratic this time
x = 4
a length cannot be −6; check 4 × 6 = 24 ✓
Different: the product rule can lead to a quadratic, and the negative root is thrown away.
An arc from the angle: two secants meet outside at 25°; the far arc is 100°. Find the near arc.
25 = 100 − n2
half the difference
50 = 100 − n, n = 50°
check: 100 − 502 = 25 ✓
Different: the angle is given and an arc is the unknown — double first, then subtract.
Two tangents: tangents from P touch the circle, and the minor arc between the points of contact is 140°. Find the angle at P.
major arc = 360° − 140° = 220°
the far arc
∠P = 220 − 1402 = 40°
half the difference
Different: both arcs come from one number. A shortcut check: the two radii make right angles with the tangents, so ∠P = 180° − 140° = 40° ✓
On the exam
Expect a missing chord, secant or tangent length, an angle made by chords, secants or tangents, and the radius-perpendicular-to-tangent fact hidden in a right triangle. The fastest route is to decide first whether the meeting point is inside, on, or outside the circle. The 10-second check: lengths use products, never sums, and an angle outside the circle is smaller than the arcs that make it.
Now you try
Practice problem
Find the tangent length
From a point outside a circle, a tangent of length t and a secant are drawn. The secant's external piece is 4 and its far piece is 5. What is t?
The whole secant is 4 + 5 = 9, so t² = 9 × 4 = 36 and t = 6. A uses the far piece instead of the whole: 5 × 4 = 20. C forgets to take the square root, leaving t². D multiplies the whole by the far piece, 9 × 5.
Hint: write the whole secant down as a number before you start —
that one step prevents three of the four wrong answers.
