CBSE Class 10 · Mathematics
Circles
Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.
This lesson follows the official chapter “Circles” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.
- CBSE Class 10
- Medium level
- 8 concepts
1Tangent to a Circle
A tangent meets a circle at exactly one point. The radius to that point is perpendicular to the tangent (as taught). Tangent is a one-point meet plus the 90° radius, not a line that “grazes” in a doodle.
A secant that cuts twice is not this tangent.
Figure. A tangent meets the circle at one point P and is perpendicular to the radius OP there. The right-angle square is the exam mark; the rim is not drawn.
How it works
- Mark the meet-pointOne point.
- Draw the radius to that pointThe radius.
- Mark 90° with the tangentThe tangent-test.
2Number of Tangents from a Point on a Circle
From a point outside, two tangents can be drawn; they are equal in length (as taught). From a point on the circle, one tangent; from inside, none in this school story. Number is a place-test, not a guess.
Drawing three pretty grazes from one outside point is a doodle.
Figure. From a point you draw 0, 1 or 2 tangents according as the point sits inside, on, or outside the circle. A chord is not a tangent — it cuts twice.
How it works
- Locate the point: outside, on, or insideThe place.
- Name two / one / none as taughtThe number.
- If two, mark them equalEqual tangents.
3A definition is a test you can run
Tangent is a test: one meet-point plus radius ⊥ tangent. If you only say “touch”, you have a heading.
A thick marker that hides a second meet is not the test.
Figure. The school test is the meet count. Exactly one contact point names a tangent; two names a chord. A pretty disk is not the test.
How it works
- Name the meet and the radiusThe object.
- Show one point and 90°The test.
- Then the word has contentThe definition ran.
4Name the given before the unknown
The given is the centre, the point, and the tangent. The unknown is a length or “how many”. Copy “outside” before you claim two.
Using a point that sits on the circle as if it were outside is a silent given.
Figure. Name the given radius and OP before chasing the tangent length. The unknown is t, from the right triangle at the contact point.
How it works
- Copy O, the point, and the lineThe given.
- Name length or the countThe unknown.
- Then 90° or the two-equalGiven first.
5One worked case is enough at this class
One tangent-pair is enough. A page of the same two-equal clone does not add a new 90°.
Ten twins of “two from outside” are still one idea.
Figure. One worked case: r = 5, OP = 13. The tangent is 12 because 5² + 12² = 13². Right angle at the contact T, not at O.
How it works
- Run one 90° or one equal-pairThe case.
- A second only as a checkOptional.
- StopThis class.
6A miss: swapping the school name for the picture
A bicycle-wheel photo is a setting. The maths is one meet and a 90° radius. A tyre is not a proof.
A coin still is not two equal tangents.
Figure. The chapter name is Circles, but the test is the perpendicular at the contact. Drawing a disk and stopping swaps the school name for the picture.
How it works
- Keep the wheel as a storyA setting.
- Mark meet, radius, 90°The maths.
- Then count from a pointDo not swap.
7Check by the opposite action or the opposite test
Check 90° by the converse (perp from centre to a line meeting at one point), or check equal tangents by walking both from the outside point to the meets.
A look-only “seems 90” is not the opposite test.
Figure. The opposite test of t² = OP² − r² is 5² + 12² against 13². 25 + 144 = 169, so the tangent length stands.
How it works
- State 90° or equal lengthsThe claim.
- Rebuild from the converse or the two walksThe check.
- Mend if they disagreeHonest.
8Keep the claim at this chapter, not the next
This chapter’s claim stops at tangent and how many. It does not steal sector-area, and it does not start a Class 9 chord-angle dump unless a mark needs it.
A page that already “computes πr² of a segment” here misses the areas chapter.
Figure. Hold the claim at tangents and the right angle at contact. Sector area and πr² wait for Areas Related to Circles.
How it works
- Keep tangent ⊥ radius and the two-from-outsideThis chapter.
- Leave sector and segment for areas related to circlesThe next map.
- Do not treat the map as already answeredClaim-size.
A tangent meets the circle at
- Exactly one point — and the radius there is 90° to it
- Two points always
- The centre
One-point meet plus the perpendicular.
Notes
- The official chapter title is “Circles”. Teach the school test for that title, not a contest shortcut.
- If a step needs a later class, stop. The next official chapter will pick it up.
Formulas
- tangent ⊥ radius at the contact point
- two tangents from an external point are equal
Recap
Hold these pegs from the official chapter “Circles”. The wording is ExamMaster’s teaching, not a textbook recap.
- Tangent to a Circle
- A tangent meets a circle at exactly one point.
- Number of Tangents from a Point on a Circle
- From a point outside, two tangents can be drawn; they are equal in length (as taught).
- A definition is a test you can run
- Tangent is a test: one meet-point plus radius ⊥ tangent.
- Name the given before the unknown
- The given is the centre, the point, and the tangent.
- One worked case is enough at this class
- One tangent-pair is enough. A page of the same two-equal clone does not add a new 90°.
- A miss: swapping the school name for the picture
- A bicycle-wheel photo is a setting. The maths is one meet and a 90° radius. A tyre is not a proof.
Practise Circles
Reading is free and needs no account. Practice, mocks and progress live in the app.
- A 4-question practice set that ends the chapter
- 1 quick check with worked explanations
- Timed mocks scored with the real marking scheme
- Readiness tracked per topic, kept on your device