A circle and a line can meet in only three ways, and which one happens is decided by comparing the distance d from the centre to the line with the radius r.
If d is less than r the line cuts the circle at two points, if d = r it just touches at one point, and if d is greater than r they do not meet.
A line touching the circle at one point, a tangent, is always perpendicular to the radius at the point of contact, so the radius, the tangent and the segment to the centre form a right triangle.
That right triangle shows both that two tangents drawn from an outside point are equal and how to find their length with Pythagoras' theorem.
How a circle and a line can meet
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A circle of radius r = 80 and a line at distance d from the centre, with the chord length, as d changes
80
📏 Deciding with d and r
①If d < r, the line meets the circle at two points; it is a secant, and the segment between the two points is a chord.
②If d = r, the line meets the circle at one point and is a tangent.
③If d > r, they do not meet.
④For a secant, the chord length is 2√(r² − d²). The radius in the picture is 80, so setting d to 80 gives a tangent, and setting d to 0 makes the chord the diameter, 160.
Properties of tangents
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Two tangents drawn from an outside point P, their points of contact A and B, and the right angles with the radii
140
✨ Two properties of tangents
①A tangent is perpendicular to the radius at the point of contact (OA ⊥ PA).
②So △OAP and △OBP are right triangles sharing the hypotenuse OP with OA = OB; they are congruent, and two tangents from an outside point are equal (PA = PB).
③By Pythagoras' theorem the tangent length is √(OP² − r²). Lengthening OP in the picture makes both tangents longer, always by the same amount.
The perpendicular bisector of a chord
Property of chords
the perpendicular from the centre to a chord bisects the chord
Conversely, the perpendicular bisector of a chord passes through the centre.
Length of a chord
chord length = 2√(r² − d²)
d is the distance from the centre to the chord.
💡 Theorems about chords
①Two chords at the same distance from the centre are equal, and conversely equal chords are equally far from the centre.
②The closer a chord is to the centre, the longer it is.
③The longest chord is the diameter, which passes through the centre (d = 0).
The angle between a tangent and a chord
Tangent–chord angle
angle between a tangent and a chord = inscribed angle on the arc inside that angle
This links to inscribed angles in the next topic.
📐 Using this property
①The angle between a tangent and a chord through the point of contact equals the inscribed angle standing on the arc inside that angle.
②So an angle on the tangent side can be moved to an inscribed angle inside the circle and calculated there.
③When moving an angle, always check that both angles stand on the same arc.
Worked examples
Example 1
In a circle of radius 5, find the length of a chord whose distance from the center is 3.
1
The perpendicular from the centre bisects the chord, so the radius 5, the distance 3 and half the chord form a right triangle.
chord = 2√(5²−3²)
2
Half the chord is 4, so the chord is twice that.
= 2√16 = 8
▸ 8
Dropping a perpendicular from the centre to a chord creates a right triangle, and Pythagoras' theorem gives the chord length.
Example 2
From an external point P, find the length of the tangent to the circle. (OP=13, radius r=5)
1
The tangent is perpendicular to the radius, so use the right triangle with hypotenuse OP.
tangent = √(13²−5²)
2
Take the square root of 13² − 5² = 169 − 25 = 144.