10. Circles

5 topics1 h

Chapter 10 of NCERT Mathematics for Class 10: Circles. Three sections — Where a line can meet a circle, The tangent and the radius and Tangents drawn from a point. Five videos, 1 h in all.

Worked answers to the 17 exercise questions in this chapter: Exercise 10.1 · Exercise 10.2

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Part 1

Where a line can meet a circle

10 minCounting shared points sorts every line into one of three kindsA line and a circle can share nothing, one point, or two, and never anything else. Three drawings are not three examples out of many - they are the complete list, and one measurement tells you which one you are looking at before you draw it.10 min · 11 parts

Part 2

The tangent and the radius

10 minA tangent as what a secant becomes when its two crossings mergeA tangent is not a second kind of line standing beside the secant. It is the secant caught at the instant its two crossings run into each other and the chord between them has run out - and the explanation watches that happen twice, once by turning and once by sliding.10 min · 11 parts14 minWhy the radius meets it at a right angle, argued from shortest distanceThe spoke of a rolling wheel looks square to the road every time you catch it - and looking square is not being square. The explanation gets the right angle out of a COUNT: a tangent shares exactly one point, so every other point of that line is farther from the centre, so the radius is the shortest reach, and the shortest reach to a line is the perpendicular one.14 min · 13 parts

Part 3

Tangents drawn from a point

12 minNone, one or two, depending on where the point sitsHow many tangents pass through a point is not a fact about tangents. It is a fact about the point - and the only thing that matters about the point is whether it sits inside the circle, on it, or outside. None, one, two. There is no fourth answer, and one expression knows all three before you draw anything.12 min · 12 parts15 minWhy the two from an outside point are always equally longThe two tangents from an outside point come out the same length, and the figure is symmetric about the line to the centre - which is exactly why the figure proves nothing. The symmetry is the conclusion, not the premise. Here is the argument instead: one theorem, spent twice, and then spent four more times.15 min · 13 parts

What you will be able to do

  • Sort a line drawn against a circle into one of exactly three kinds by counting the points the two share
  • State what a tangent is by the count of points it shares with the circle, and restate it as a secant whose chord has length zero
  • State what Theorem 10.1 claims, distinguishing its hypothesis from its conclusion
  • Classify a chosen point as inside, on, or outside a given circle by comparing its distance from the centre with the radius
  • State what Theorem 10.2 claims and identify the two triangles its proof compares

What this chapter assumes you already know

  • A circle as the set of points at one fixed distance from a fixed centre, and the words centre, radius, chord, arc, segment and sector — all carried over from Class IX and assumed by name on the chapter's opening page
  • That the distance from a point to a line means the perpendicular distance
  • Pythagoras' theorem in a right triangle
  • That two points determine exactly one straight line
  • That a chord is the segment joining the two points a secant shares with the circle, and that its length is a number that can shrink
  • That parallel lines all point the same way, so sliding one across a circle changes only its distance from the centre
  • The idea of a quantity approaching a value as a position is varied continuously
  • That every point of the circle is at distance r from the centre, points inside at less than r, points outside at more

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "A tangent is a line that touches but does not cross."
  • "A chord and a secant are the same thing."
  • "A line very close to the circle is almost a tangent."
  • "Non-intersecting means parallel."
  • "Which case you get depends on how the line is tilted."
  • "There are three kinds of line, so a line has a kind."
  • "The tangent is what you get at the end of the turning."
  • "The chord gets small, so the two crossings get close, so eventually they are close enough."