Bihar Board Class 10 · Maths · Matric exam 2027
Chapter 10: Circles — notes
The chapter’s 5 parts in the book’s order — the first idea of each and the points Medha Ma’am writes on the board — and the summary at the end. The full lesson and a check question for each part are in the app.
1. A circle and a line: three situations
Okay, so you already know what a circle is.
- • Circle: all points at constant distance (radius) from centre
- • Line PQ and circle: three cases
- • No common point -> non-intersecting line
- • Two common points -> secant
- • One common point -> tangent
2. What a tangent is, and that there is only one at a point
Okay, so let's talk about tangents.
- • Tangent: touches circle at only one point
- • That point = point of contact
- • At one point of a circle, only ONE tangent exists
- • Tangent = secant whose two meeting points coincide
3. Tangent is perpendicular to the radius (Theorem 10.1)
Okay, so here's an important property.
- • Theorem 10.1: tangent perpendicular to radius at point of contact
- • Circle centre O, tangent XY at P
- • Q any other point on XY: Q lies outside circle
- • So OQ > OP for every such Q
- • Shortest distance OP -> OP is perpendicular to XY
4. Tangents from a point: inside, on, outside the circle
Okay, so let's see how many tangents we can draw from a point.
- • Point inside circle: no tangent possible
- • Point on circle: exactly one tangent
- • Point outside circle: exactly two tangents
- • PT1, PT2 = lengths of tangents from external point P
5. Lengths of tangents from an external point are equal (Theorem 10.2)
Okay, so here's another key fact about circles.
- • Theorem 10.2: tangents from external point are equal
- • PQ = PR for tangents from P
- • Proof: triangle OQP cong triangle ORP (RHS), OQ = OR, OP common
- • Chord PQ = 8cm, radius = 5cm, PR = RQ = 4cm, OR = 3cm
- • TP/PO = RP/RO -> TP/5 = 4/3 -> TP = 20/3 cm
Summary
Okay, so let's look back at everything in this chapter. See, a line can meet a circle in three ways. It can miss the circle completely — that's a non intersecting line. It can cut the circle at two points — that's a secant. Or it can just touch the circle at one point — that's a tangent, and that single point is called the point of contact. Now here's something nice — a tangent is really a secant whose two meeting points have come together into one. And at any single point of a circle, only one tangent can be drawn there. Then we proved Theorem 10.1 — the tangent at any point of a circle is always perpendicular to the radius drawn to that point of contact. After that, we looked at where a point sits. A point inside the circle — no tangent from it. A point on the circle — exactly one tangent. A point outside the circle — exactly two tangents. Finally, Theorem 10.2 told us something useful — the two tangent lengths drawn from an external point to a circle are always equal. We proved it using two congruent right triangles, formed with the radii and the common tangent segment, just like in our chord and tangent example. Now, tell me — out of all this, which idea did you find the most surprising?
AskMedha · Medha Ma’am — Bihar Board Class 10 Maths, Chapter 10. Written from the textbook, checked twice.
Questions parents and students ask
- How do I get the "Circles" notes as a PDF?
- Press "Save as PDF" — the phone’s print sheet opens; choose "Save as PDF" there. The PDF holds the notes only.
- Where are these notes written from?
- From the chapter’s own textbook text, part by part, in the book’s own terms; a second, independent model then checked every part against the book. The full lesson, a check question for each part and the practice are in the AskMedha app.
The same preparation, with Medha Ma’am — on your phone
Every question of every chapter with its model answer, practice questions in the board’s own shapes, 60 minutes of class a day, written answers marked, a test every Sunday and a WhatsApp report to parents.
Medha Ma’am is an AI teacher, not a live tutor. Every answer is checked twice; a person takes responsibility.