None, one or two, depending on where the point sits

What to assume they know

• Counting shared points sorts every line into one of three kinds — the three positions of a line against a circle, sorted by shared points
• Why the radius meets it at a right angle, argued from shortest distance — Theorem 10.1, and Remark 1's uniqueness of the tangent at a point of the circle
• That a point is inside, on, or outside the circle according as its distance from the centre is less than, equal to, or greater than the radius
• Pythagoras' theorem
• That a segment has a length and a line does not

What they should be able to do

• Classify a chosen point as inside, on, or outside a given circle by comparing its distance from the centre with the radius
• State how many tangents pass through the point in each of the three cases
• Explain why no line through an interior point can be a tangent
• Explain why the "exactly one" of the middle case is already proved rather than merely observed
• Identify the points of contact of the two tangents from an external point
• Distinguish the tangent line from the length of the tangent from a stated point
• Compute the length of the tangent from the radius and the distance from the centre
• Read the three cases off a single expression and say what its arithmetic does in each case

Where it usually goes wrong

• "You can always draw two tangents from a point." Only from outside. The count is a property of where the point sits, and the chapter's three cases exist precisely to stop this generalisation.
• "From a point on the circle you get two tangents — one going each way." Those two directions are the two halves of a single line. One line, one tangent. Remark 1 on p. 147 is what closes this off.
• "From inside you can at least reach the far side of the circle." Every line through an interior point exits through the circle twice. There is no far side to reach without crossing.
• "The length of the tangent is the length of the tangent line." A line has no length. The phrase names the segment from the stated external point to the point of contact, and it changes if you change the point.
• "The length of the tangent is how far the point is from the circle." It is not — 25 cm out from the centre of a 7 cm circle puts you 18 cm from the nearest point of the circle and 24 cm from the point of contact. The tangent runs sideways, not straight in.
• "Add the squares to get the tangent length." The distance to the centre is the hypotenuse, because the right angle sits at the point of contact. Subtract.
• "There might be a third tangent from an external point if you hunt hard enough." There cannot be: the points of contact are pinned to a second circle, and two circles cannot share three points.
• "Case 2 is obvious from Case 3 by moving the point inwards." Moving P inwards makes the two tangents swing together, but the limit argument is not the proof — the uniqueness at a point of the circle comes from the perpendicular being unique.

Questions to check understanding

• State how many tangents can be drawn from a point inside, on, and outside a circle
• Given a radius and a distance from the centre, find the length of the tangent
• Given a tangent length and a distance from the centre, find the radius, and the reverse
• Multiple choice on the radius from a tangent length and a distance to the centre
• Short-answer: explain why no tangent can pass through a point inside a circle
• Draw the tangents from a stated external point and mark the points of contact
• Explain the difference between the tangent at a point of the circle and the length of the tangent from an external point

Examples worth working on the board

Inputs only. Values marked verified are worked out here on the chapter's printed data.

• Activity 3, the instruction (§10.3, pp. 147–148). Draw a circle. Put a point P inside it and try for a tangent through P. Then put P on the circle and try again. Then put P outside and try again. Three attempts, three different answers.
• Fig. 10.6 (i) (p. 148). Read from the printed page: a circle with a point labelled P inside it, towards the upper right, and three arrow-headed lines drawn through P, each entering and leaving the circle — six crossings in all across the three lines. The drawing's message is that the failure is not a matter of aim.
• Why no interior point can work. This argument is added here; the chapter reports the outcome of the activity and gives no reason. If P is inside then its distance from the centre is less than the radius. Any line through P stands some perpendicular distance from the centre — call it p, keeping d free for the centre-to-P distance this brief uses later — and p cannot exceed the length OP, so p is also less than the radius. A line at less than the radius from the centre meets the circle twice. So every line through P is a secant, with no exceptions to chase.
• Fig. 10.6 (ii) (p. 148). Read from the printed page: a circle with P marked on the circle itself, on the right-hand side, and a single arrow-headed line drawn through P, running clear of the circle on both sides of it.
• What Case 2 really rests on. The chapter says there is one and only one tangent at a point of the circle. The existence half came out of Activity 1 in the previous module; the uniqueness half is Remark 1 on p. 147, which reads it off Theorem 10.1 — at P there is only one line perpendicular to the radius OP.
• Fig. 10.6 (iii) (p. 148). Read from the printed page: a circle on the left with an external point P to its right; two straight lines run from P back to the circle, touching it at points labelled T₁ above and T₂ below, and continuing past the touching points with arrowheads at their far ends. The arrowheads matter: the lines run on forever, and only the pieces PT₁ and PT₂ are what the chapter is about to call lengths.
• Why exactly two from an external point. This argument is added here; the chapter observes the count and does not justify it. If PT is a tangent touching at T, then by Theorem 10.1 the angle at T between TP and TO is a right angle — so T lies on the circle having OP as a diameter. The touching points are therefore the common points of two circles: the original one, and the circle on OP as diameter. Two distinct circles share at most two points, which caps the count at two; and when P is outside, the second circle reaches from O out past the original circle's rim, so it genuinely crosses it and both points exist. Hence exactly two, never three.
• The definition that Case 3 makes possible (p. 148). The chapter defines the length of the tangent from P as the distance along the tangent from P to the point of contact. Note carefully what this is not: it is not the distance from P to the circle, and it is not a property of the line.
• Computing that length. Derived, not printed: with r the radius and d the distance from P to the centre, the right angle at the point of contact gives d² = r² + (tangent length)², so the tangent length is √(d² − r²). The chapter never writes this formula down, but every numerical question in Exercise 10.2 is an instance of it.
• The formula reproduces all three cases. Verified as algebra: with r = 5, take d = 3 and d² − r² = −16, which has no square root — no tangent, matching Case 1; take d = 5 and the length is 0, the degenerate case where P is its own point of contact, matching Case 2; take d = 13 and the length is √(169 − 25) = 12, two tangents of that length, matching Case 3. One expression, three verdicts. This unification is added here.
• Exercise 10.2 question 1, as data (p. 151 — the exercise sits two pages on but is the natural drill for this topic). Tangent length 24 cm, distance from the point to the centre 25 cm; options 7 cm, 12 cm, 15 cm, 24.5 cm. Verified: the radius is √(625 − 576) = 7 cm.
• Exercise 10.2 question 6, as data (p. 152). A point 5 cm from the centre, tangent length 4 cm, radius wanted. Verified: √(25 − 16) = 3 cm.
• A check worth doing. For the 25 cm / 24 cm / 7 cm triple, the nearest point of the circle to P is only 25 − 7 = 18 cm away, while the tangent length is 24 cm. Verified. The tangent length is therefore not the distance from P to the circle, and the numbers make the point far better than a warning does.
• The open question this topic hands on. The p. 148 discussion — not §10.3, which runs on to p. 152 — closes by asking the student to measure PT₁ and PT₂ and notice that they agree. Deliberately leave the explanation there: the measurement is an invitation, and the proof is the next topic.

Figures to have open

• Fig. 10.6's three panels (p. 148) redrawn as a single stacked strip, so the three cases are read against each other rather than one page-scroll apart. The chapter's own figure; keep P labelled in all three and keep the arrowheads that say these are lines.
• The two-circle picture: the given circle, and the circle on OP as a diameter, crossing it at exactly T₁ and T₂. Standard schematic and entirely not in the book — the chapter draws nothing like it, and this is the picture that answers "why two".
• The right triangle O, T, P with the right angle at T, labelled r, d and the tangent length. Standard schematic.
• A number line or slider for d running from 0 through r and out beyond, with the tangent count reading 0, 1, 2 as it passes. Standard schematic, not in the book.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, Chapter 10 "Circles", §10.3 Number of Tangents from a Point on a Circle — the section opens on p. 147 and runs to p. 152, and this topic takes its first two pages, 147–148: Activity 3, Fig. 10.6 (i)–(iii), Cases 1 to 3, the definition of the length of the tangent, and the invitation to measure PT₁ and PT₂ that closes the p. 148 discussion
• Remark 1 after Theorem 10.1, p. 147, which supplies the uniqueness in Case 2
• Exercise 10.2 questions 1 and 6, pp. 151–152, used here as the numerical drill for the tangent length; these sit inside the next topic's page range and are named as a deliberate forward reference
• The chapter summary, §10.4, p. 153, whose third point is what the closing measurement anticipates