Bihar Board Class 10 · Maths · Matric exam 2027
Chapter 11: Constructions — notes
The chapter’s 6 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. Dividing a Line Segment in a Given Ratio
Okay, so — you already know how to bisect a line, or draw a perpendicular bisector, with a ruler and compass.
- • Given AB, divide in ratio m:n (here 3:2)
- • Draw ray AX, acute angle with AB
- • Mark 5 (=m+n) equal points A1...A5 on AX
- • Join BA5
- • Through A3 draw line parallel to A5B, meeting AB at C
- • AC : CB = 3 : 2
2. Alternative Method for Dividing a Line Segment
Okay, so there's another way to divide AB in the ratio 3 : 2.
- • Draw ray AX, acute angle with AB
- • Draw ray BY parallel to AX, angle ABY = angle BAX
- • Mark A1,A2,A3 on AX and B1,B2 on BY, all equal lengths
- • Join A3B2, meeting AB at C
- • Triangle AA3C ~ Triangle BB2C
- • AC : BC = AA3 : BB2 = 3 : 2
3. Constructing a Smaller Similar Triangle (Scale Factor Less Than 1)
Okay, so now let's build a smaller triangle, similar to triangle ABC.
- • Example: sides = 3/4 of triangle ABC's sides
- • Draw ray BX, acute angle with BC, opposite side from A
- • Mark 4 equal points B1,B2,B3,B4 on BX
- • Join B4C; through B3 draw line parallel to B4C, meeting BC at C'
- • Through C' draw line parallel to CA, meeting BA at A'
- • Triangle A'BC' is required, A'B/AB = BC'/BC = 3/4
4. Constructing a Larger Similar Triangle (Scale Factor More Than 1)
Okay, so sometimes we need the new triangle to be bigger than the given one.
- • Example: sides = 5/3 of triangle ABC's sides
- • Mark 5 equal points B1...B5 on ray BX
- • Join B3 to C; through B5 draw line parallel to B3C
- • This meets extended BC at C'
- • Through C' draw line parallel to CA, meeting extended BA at A'
- • Triangle A'BC' is required, sides = 5/3 of ABC's sides
5. Drawing a Tangent at a Point on a Circle
Okay, so let's talk about a point on a circle.
- • Point inside circle: no tangent possible
- • Point on circle: exactly one tangent
- • Tangent at a point is perpendicular to radius through it
- • Step: draw radius through the point
- • Step: draw perpendicular to radius at that point = tangent
6. Constructing Tangents from a Point Outside the Circle
Okay, so, suppose P is a point outside the circle.
- • Given: circle with centre O, point P outside it
- • Join PO, find midpoint M
- • With M as centre, MO as radius, draw a circle
- • This circle cuts given circle at Q and R
- • Join PQ and PR = required tangents
- • Angle PQO = 90° (angle in semicircle), so PQ, PR are tangents
Summary
Okay, so let's look back at everything in this chapter. You built on your Class IX work with ruler and compass, and learnt three main constructions. First, you learnt to divide a line segment in a given ratio, like 3:2. You did this using a ray with equal marked points, and you saw two different ways to do it. Both of these were justified using similar triangles and the Basic Proportionality Theorem. Then, using this same idea, you learnt to construct a triangle similar to a given triangle, for any scale factor. The new triangle could be smaller than the original — say, 3/4 of its sides — or larger, say 5/3 of its sides. After that, you learnt how to draw tangents to a circle. When the point lies on the circle, there's only one tangent at it, and it's perpendicular to the radius there. When the point lies outside the circle, there are two tangents from it. You found these using a circle drawn through the midpoint of PO, and the fact that the angle in a semicircle is 90°. And notice, every single construction you did came with its own mathematical reasoning. It wasn't just steps — each one showed you exactly why it gives the correct result. So tell me, out of these three constructions, which one did you find the most interesting?
AskMedha · Medha Ma’am — Bihar Board Class 10 Maths, Chapter 11. Written from the textbook, checked twice.
Questions parents and students ask
- How do I get the "Constructions" 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.
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