What is a circle? A circle is a collection of all points in a flat plane that are at a constant, fixed distance from a central fixed point. Think of a giant Ferris wheel spinning around its central axle, a round dinner plate, or the boundary of a circular coin. The path traced by these boundaries represents a perfect mathematical circle.
Let us review the foundational parts of a circle that you must remember:
- Centre: The fixed point in the exact middle of the circle.
- Radius: The straight line segment connecting the centre to any point on the boundary.
- Diameter: A straight line passing straight through the centre, connecting two opposite points on the boundary. It is exactly twice the length of the radius.
- Chord: A straight line segment connecting any two points on the circle’s boundary. The diameter is simply the longest possible chord!
- Arc: A continuous portion or segment of the circular boundary line.
- Circumference: The total perimeter or outer boundary distance around the entire circle.
Regions of a Circle. A circle divides a flat plane into three distinct parts: (1) the interior — the space trapped inside the circular boundary; (2) the exterior — the entire infinite space outside the boundary line; and (3) the circle itself — the boundary line where every point lies at an exact radius distance from the centre.
Lines associated with a circle. When a straight line and a circle share a plane, exactly three situations can arise, and the deciding factor is the perpendicular distance $d$ from the centre to the line compared with the radius $r$:
- Non-intersecting line ($d>r$): The line passes completely outside the circle and shares no point with it.
- Secant ($dtwo distinct points. The portion of a secant lying inside the circle is a chord.
- Tangent ($d=r$): A special line that just brushes the circle, touching it at exactly one point. That single spot is the point of contact.
Tangent as the limiting position of a secant. Picture a secant $PQ$ cutting the circle at two points. Slide one point along the boundary towards the other; the chord shrinks. When the two points finally coincide, the secant becomes a tangent. This is exactly why a tangent meets the circle at one point only — it is the limiting case where the two intersection points of a secant merge into one. This idea is the foundation for everything that follows in this chapter.
Number of tangents from a point. How many tangents you can draw depends on where the point sits relative to the circle: from a point inside the circle you can draw 0 tangents (every line through an interior point is a secant); from a point on the circle exactly 1 tangent; and from a point outside the circle exactly 2 tangents.
| Line Type | Distance vs radius | Shared Points | Relationship |
|---|
| Non-intersecting | $d>r$ | 0 | Passes outside the boundary |
| Secant | $d| 2 | Cuts the interior, forms a chord | |
| Tangent | $d=r$ | 1 | Skims one point, the point of contact |
The word “tangent” comes from the Latin tangere, “to touch”, while “secant” comes from secare, “to cut”. The names literally describe the behaviour: one touches, the other cuts.
Common mistakes to avoid. (i) A tangent is not simply “a short line near the circle” — the defining feature is exactly one common point. (ii) The deciding distance $d$ must be the perpendicular distance from the centre to the line, never a slanted distance. (iii) A diameter is the longest chord, but a chord is not always a diameter — only the chord passing through the centre is. (iv) “Two tangents from an external point” does not mean two points of contact on the same line; it means two different tangent lines, each with its own point of contact.
A straight line lies in the plane of a circle. The perpendicular distance from the centre to the line equals the radius of the circle. Classify the line as non-intersecting, secant or tangent.
Solution- Step 1: Compare the perpendicular distance $d$ with the radius $r$.
- Step 2: We are told $d=r$.
- Step 3: A line with $dr$ is non-intersecting, and with $d=r$ touches the circle at exactly one point.
- Step 4: Since $d=r$, the line meets the circle at exactly one point.
Answer: The line is a tangent.
A circle has diameter $16$ cm. A line $L$ in its plane is at a perpendicular distance of $6$ cm from the centre. How many points does $L$ share with the circle, and what is it called?
Solution- Step 1: Radius $r=\dfrac{16}{2}=8$ cm.
- Step 2: The given distance is $d=6$ cm.
- Step 3: Since $d=6<8=r$, the line passes through the interior.
- Step 4: A line through the interior cuts the boundary at two points.
Answer: $2$ points; the line is a secant.
A straight wooden strip touches a circular clock at exactly one point. The shortest distance from the clock’s centre to the strip is $15$ cm. Find the circumference of the clock. (Use $\pi=\tfrac{22}{7}$.)
Solution- Step 1: Touching at exactly one point means the strip is a tangent, so the shortest (perpendicular) distance equals the radius: $r=15$ cm.
- Step 2: Circumference $C = 2\pi r = 2\times\dfrac{22}{7}\times 15$.
- Step 3: $C = \dfrac{2\times 22\times 15}{7} = \dfrac{660}{7}$ cm.
- Step 4: $\dfrac{660}{7}\approx 94.29$ cm.
Answer: $\dfrac{660}{7}$ cm $\approx 94.29$ cm.
How many tangents can be drawn to a circle from (i) a point inside it, (ii) a point on it, (iii) a point outside it?
Solution- Step 1: From an interior point every line cuts the circle twice, so no line can touch at one point: $0$ tangents.
- Step 2: From a point on the circle there is one line perpendicular to the radius there: $1$ tangent.
- Step 3: From an external point two distinct tangents can be drawn: $2$ tangents.
Answer: (i) $0$, (ii) $1$, (iii) $2$.
A line $L$ is at perpendicular distance $10$ cm from the centre of a circle of radius $7$ cm. State whether $L$ meets the circle.
Solution- Step 1: Compare $d=10$ cm with $r=7$ cm.
- Step 2: Since $d=10>7=r$, the line stays entirely outside the circle.
- Step 3: It shares no point with the circle.
Answer: It does not meet the circle (non-intersecting line).
The longest chord of a circle is $26$ cm. A second chord is drawn at a perpendicular distance of $5$ cm from the centre. Find the length of this second chord.
Solution- Step 1: The longest chord is the diameter, so diameter $=26$ cm and radius $r=13$ cm.
- Step 2: The perpendicular from the centre bisects the chord. Half-chord $=\sqrt{r^2-d^2}=\sqrt{13^2-5^2}$.
- Step 3: $=\sqrt{169-25}=\sqrt{144}=12$ cm.
- Step 4: Full chord $=2\times 12 = 24$ cm.
Answer: $24$ cm.
How many tangents can be drawn to a circle that are parallel to a given line? Justify briefly.
Solution- Step 1: A tangent parallel to a given direction touches the circle where the radius is perpendicular to that direction.
- Step 2: There are exactly two such diametrically opposite points on the circle.
- Step 3: Hence exactly two parallel tangents exist for any given direction.
Answer: Exactly $2$ tangents.
Two concentric circles have radii $5$ cm and $3$ cm. A chord of the larger circle touches the smaller circle. Find the length of this chord.
Solution- Step 1: The chord touches the inner circle, so the radius of the inner circle ($3$ cm) is perpendicular to the chord at the contact point and bisects it.
- Step 2: This $3$ cm radius and the larger radius $5$ cm form a right triangle with the half-chord.
- Step 3: Half-chord $=\sqrt{5^2-3^2}=\sqrt{25-9}=\sqrt{16}=4$ cm.
- Step 4: Full chord $=2\times 4 = 8$ cm.
Answer: $8$ cm.
A point $P$ is at distance $13$ cm from the centre of a circle. The length of the tangent from $P$ is $12$ cm. Find the radius.
Solution- Step 1: For a tangent, $OP^2 = r^2 + (\text{tangent length})^2$, with $OP$ the hypotenuse.
- Step 2: $13^2 = r^2 + 12^2 \Rightarrow 169 = r^2 + 144$.
- Step 3: $r^2 = 169-144 = 25$.
- Step 4: $r = \sqrt{25} = 5$ cm.
Answer: Radius $=5$ cm.
How many common tangents can be drawn to two circles that touch each other (i) externally and (ii) internally?
Solution- Step 1: When two circles touch externally, they have two direct (external) tangents and one tangent at the common point: $3$ in all.
- Step 2: When two circles touch internally, the only common tangent is the one at the point of contact: $1$.
- Step 3: State both counts.
Answer: (i) $3$ common tangents; (ii) $1$ common tangent.
Prove that the line drawn through the centre of a circle perpendicular to a tangent passes through the point of contact.
Solution- Step 1: Let the tangent touch the circle at $P$, centre $O$. By the tangent theorem, $OP\perp$ tangent.
- Step 2: From $O$, only one perpendicular can be dropped to a given line, and it meets the line at a unique foot.
- Step 3: Since $OP$ is perpendicular to the tangent and $P$ lies on the tangent, $P$ is that unique foot.
- Step 4: Therefore the perpendicular from the centre to the tangent passes through the point of contact $P$.
Answer: Proved: the perpendicular through $O$ meets the tangent at the point of contact.
A circle of radius $r$ has a chord at distance $d$ from the centre. Write the chord length, then evaluate it for $r=10$ cm and $d=6$ cm.
Solution- Step 1: The perpendicular from the centre bisects the chord; half-chord $=\sqrt{r^2-d^2}$.
- Step 2: Chord length $=2\sqrt{r^2-d^2}$.
- Step 3: Substitute: $2\sqrt{10^2-6^2}=2\sqrt{100-36}=2\sqrt{64}$.
- Step 4: $=2\times 8 = 16$ cm.
Answer: Chord $=2\sqrt{r^2-d^2}=16$ cm.