Chapter 9.1 — Introduction to Tangents and Secants
Introduction of tangent and secant of a circle. This is Lesson 1 of 4 in Chapter 9: Tangents and Secants to a Circle.
Three Ways a Line Can Meet a Circle
A straight line drawn across a circle only ever does one of three things: cross it twice, touch it exactly once, or miss it entirely. This chapter is built almost entirely around the middle case — the line that touches a circle at exactly one point — and everything that single point of contact forces to be true.
Secant, Tangent, and Neither
- A secant is a line that intersects a circle at two distinct points — think of it as a line passing straight through the circle's interior, entering at one point and exiting at another. Every chord of a circle sits on some secant line, extended infinitely in both directions past the two points where it actually meets the circle.
- A tangent is a line that touches a circle at exactly one point. That single shared point is called the point of contact, and it's the only point the tangent line and the circle have in common — everywhere else along its entire length, the tangent line stays strictly outside the circle.
- A non-intersecting line misses the circle entirely, sharing no points with it at all. This is the least interesting case geometrically, but it's still worth naming explicitly, since a problem describing "a line and a circle" could mean any of these three relationships until the actual distance between the two is checked directly.
A line's relationship to a circle isn't fixed — the same line drawn at different distances from the centre passes through all three states in turn: two intersection points while it's still cutting through the circle, exactly one point at the precise moment it grazes the edge, then zero points once it moves past. A tangent is really that one exact instant balanced between "still crossing" and "no longer touching."
Circles and the lines that meet them show up constantly outside a geometry textbook too — a bicycle wheel touching the road, a ball resting against a wall, the edge of a coin balanced on a ruler. In every one of those cases, the touching line and the circle share exactly one point, which is precisely the situation this chapter formalises. A rolling wheel is a genuinely useful mental picture to keep throughout this chapter: the road is always tangent to the wheel, touching it at exactly one point no matter how far the wheel rolls.
It's also worth being precise about what "touches at one point" actually rules out. A line can pass extremely close to a circle without touching it — grazing near the edge, missing by a hair's width — and it's still classified as non-intersecting as long as it shares zero points with the circle. Distance, not visual closeness, is what actually decides which of the three categories a line falls into: compare the perpendicular distance from the centre to the line against the radius, and the comparison settles the question exactly (distance less than radius means secant, equal means tangent, greater means non-intersecting). This distance-versus-radius comparison is worth keeping in mind as a mental shortcut, since it settles the classification without needing to actually find where, or whether, the line and circle intersect.
The Tangent-Radius Right Angle
Every tangent obeys one fixed geometric rule, and this entire chapter leans on it repeatedly: the tangent at any point of a circle is always perpendicular to the radius drawn to that point of contact.
This makes intuitive sense once you consider the alternative: if the radius met the tangent at any angle other than 90°, the tangent line would have to dip slightly closer to the centre somewhere nearby — which would mean it cuts into the circle's interior rather than just grazing it, making it a secant instead of a tangent. The perpendicular radius is the one angle that keeps the entire line exactly on the circle's boundary at that single point and nowhere closer. Put differently, the radius to the point of contact is always the shortest possible distance from the centre to the tangent line, and the shortest distance from a point to a line is always measured along a perpendicular — so the 90° angle isn't an extra fact bolted onto tangents, it falls directly out of what "touching but not crossing" actually requires geometrically.
Finding the Length of a Tangent
Since the radius OA and the tangent PA meet at exactly 90° at the point of contact, triangle OAP is always a right triangle — which means Pythagoras' theorem finds the tangent's length directly, given only the radius and the distance from the external point to the centre. This is the entire reason the tangent-radius right angle matters practically, not just theoretically: without a guaranteed right angle somewhere in the figure, there would be no clean way to relate the tangent's length to the other two measurements at all, no matter how the problem happened to be phrased.
OP² = OA² + PA² ⟹ PA = √(OP² − OA²)Nothing about this formula is specific to circles beyond the one fact that makes it work in the first place — the guaranteed right angle at the point of contact. Every tangent-length problem in this chapter's exercises reduces to this single substitution.
A quick illustration: a circle of radius 6 cm has its centre 10 cm from an external point P. The tangent length is √(10²−6²) = √(100−36) = √64 = 8 cm. Notice that 6, 8, and 10 form a scaled-up 3-4-5 right triangle — a pattern worth watching for, since tangent-length problems are frequently built around one of the well-known Pythagorean triples rather than an arbitrary decimal, keeping the final square root clean instead of leaving an unsimplified surd.
Where the Chapter Goes From Here
Exercise 9.1 applies this tangent-length formula directly and proves a first fact about tangents drawn at a diameter's two ends. Exercise 9.2 turns to how many tangents a single external point can send toward a circle, proves those tangents always have equal length, and constructs them directly with compass and straightedge. Exercise 9.3 shifts focus entirely — from tangents to the secant's chord, and the areas the two curved regions on either side of that chord enclose. Every one of those three exercises still rests on the same two facts established here: the tangent-radius right angle, and the Pythagoras-based tangent-length formula it makes possible.
A quadrilateral formed by two tangents and two radii from a circle's centre is worth noticing early, since it resurfaces constantly in the exercises ahead: with both tangent-radius angles fixed at 90°, only the angle at the centre and the angle at the external point are free to vary — and because a quadrilateral's four angles always sum to 360°, those two remaining angles are locked into always adding up to exactly 180° between them. That single relationship, quietly built into every tangent-pair figure from the two guaranteed right angles alone, is what several problems in Exercise 9.2 lean on to find an unknown angle without any additional given information.