Reflection Class 10 ICSE — Bansal Chapter 12
Reflection Class 10 ICSE is the chapter where co-ordinate geometry turns visual. In Bansal’s Class 10 Mathematics book, chapter 12 shows you how to locate a point with two co-ordinates, then flip it across the x-axis, the y-axis or the origin. Every section below follows the book’s own order, with the rules, worked patterns and the chapter PDF for revision. This is the page to keep open while you work through the chapter.
Download Reflection Class 10 ICSE Mathematics Chapter 12 PDF
Download the Reflection Class 10 ICSE Mathematics chapter PDF from the Bansal textbook to revise offline or print for practice. It is the full chapter as printed in the book — 10 pages, 9 sections, 36 questions.
What is inside Chapter 12
10 pages · 9 sections · 4 worked examples · 36 exercise questions
| Section | Title |
|---|---|
| 12.1 | Introduction |
| 12.2 | Co-ordinate Axes |
| 12.3 | Co-ordinates |
| 12.4 | Reflection |
| 12.5 | Reflection in the line y = 0 i.e. in the x-axis |
| 12.6 | Reflection in the line x = 0 i.e. in the y-axis |
| 12.7 | Reflection in the origin |
| 12.8 | Invariant Point |
| 12.9 | Using Graph Paper |
- EXERCISE 12(A): 19 questions
- EXERCISE 12(B): 17 questions
12.1 Introduction
Co-ordinate geometry is the branch of geometry that fixes a point’s position in a plane using two numbers, called co-ordinates. That pair of numbers carries the whole idea behind this ICSE chapter. Read the point correctly, and you can place it, move it, flip it and measure between points.
The two numbers work like an address. One number tells how far the point sits sideways; the other tells how far it sits up or down. Together they pin the point to one unique spot on a flat surface, so no two points share the same pair.
Reflection needs this language because a mirror image is a point moved to a matching position on the other side of a line. To describe the new position, you use the same co-ordinate pair. That is why the chapter opens with co-ordinates before it touches reflection at all. If you want the chapters that follow, the Class 10 ICSE Mathematics hub lists them in order.
12.2 Co-ordinate Axes
The axes are the two reference lines that make co-ordinates possible. They are number lines set at right angles to each other, crossing exactly at their zeroes. The book calls them the rectangular axes, the co-ordinate axes or the axes of reference.
The horizontal number line XOX’ is the x-axis. The vertical number line YOY’ is the y-axis. Their point of intersection, O, is the origin. The word ‘rectangular’ appears because the two lines meet at a right angle, like the corner of a rectangle.
Both axes are real number lines, so they carry positive and negative values. To the right of O, the x-axis runs positive; to the left, negative. Above O, the y-axis runs positive; below, negative. That split lets a co-ordinate pair point to any spot in the plane.
Every reflection rule in this chapter is measured from these two lines, so their direction matters more than any other detail. Once you can name which line is which, the sign rules that follow make sense.
12.3 Co-ordinates
A point’s co-ordinates are a pair of numbers, written (x, y). The first number, x, is the abscissa. It measures the point’s distance from the y-axis. The second number, y, is the ordinate. It measures the point’s distance from the x-axis.
Take the point A(4, -2). Its abscissa is 4, so A sits 4 units from the y-axis. Its ordinate is -2, so A sits 2 units below the x-axis. The negative sign in the ordinate is the clue that the point lies below the horizontal axis.
Four rules in the book fix how co-ordinates are written:
- The abscissa always comes first, so a point is always stated as (x, y).
- The origin O has co-ordinates (0, 0).
- A point on the x-axis has co-ordinates (x, 0).
- A point on the y-axis has co-ordinates (0, y).
When a point sits on an axis, one of its two distances is zero. That is why (x, 0) and (0, y) carry a zero in the right slot. Get this order right now, because every reflection rule in this chapter depends on it.
12.4 Reflection
Reflection is the mirror-image transformation. When you place an object in front of a plane mirror, its image appears at the same distance behind the mirror as the object is in front. The chapter uses this exact model for points.
To find the image of a point P in a line AB, treat AB as the mirror. Find a point P’ on the other side of AB, at the same distance from AB as P. Then P’ is the image of P, and AB is the mirror line. Because P and P’ are equidistant from AB and on opposite sides, AB is also the perpendicular bisector of the segment PP’.
The transformation that maps P to P’ is called reflection. The book denotes it as M_l, where M stands for reflection and l is the line or point of reflection. So M_x means reflection in the x-axis, M_y means reflection in the y-axis, and M_o means reflection in the origin. These three symbols are the language of the rest of the chapter.
12.5 Reflection in the line y = 0 i.e. in the x-axis
The line y = 0 is simply the x-axis. Every point on the x-axis has ordinate 0, so the equation y = 0 names that whole line. When you reflect a point in the x-axis, only its ordinate changes sign.
M_x(x, y) = (x, -y)
For example, M_x(7, -4) = (7, 4). The abscissa 7 stays untouched; the ordinate -4 flips to 4.
The common mistake here is changing the wrong co-ordinate. Some students flip the abscissa out of habit. That is wrong: reflecting in the x-axis never touches the x-co-ordinate, because the point moves straight across the horizontal line.
A simple mnemonic keeps this straight: the axis you cross is the one you keep. Cross the x-axis, keep x. Cross the y-axis, keep y. Apply that one line and the sign rules stop feeling random.
12.6 Reflection in the line x = 0 i.e. in the y-axis
The line x = 0 is the y-axis. Every point on the y-axis has abscissa 0, which is why x = 0 names the whole line. Reflecting in the y-axis changes only the abscissa’s sign.
M_y(x, y) = (-x, y)
For example, M_y(-3, 8) = (3, 8). The ordinate 8 stays; the abscissa -3 flips to 3.
Compare this with the x-axis rule and the pattern is clean. M_x flips the ordinate; M_y flips the abscissa. Each axis flips the co-ordinate named after the other. That matches the mnemonic: cross the y-axis, keep y.
Watch the sign when the point already sits on the axis. A point like (0, 5) reflected in the y-axis stays (0, 5), because its abscissa is already zero. Flipping zero changes nothing.
12.7 Reflection in the origin
Reflection in the origin is different from the two axis rules because both signs change at once.
M_o(x, y) = (-x, -y)
The origin O sits between the point and its image, and the image lands exactly opposite. The point’s distance from O is preserved, but it finishes in the diagonally opposite quadrant.
Combined reflections follow a useful pattern. Work through M_x then M_y on (5, -2).
Step 1: Apply M_x first. Only the ordinate changes sign, so M_x(5, -2) = (5, 2).
Step 2: Apply M_y to the result. Only the abscissa changes sign, so M_y(5, 2) = (-5, 2).
Step 3: Repeat in the reverse order. M_y(5, -2) = (-5, -2), then M_x(-5, -2) = (-5, 2).
Final answer: Both orders give (-5, 2), so M_x \cdot M_y = M_y \cdot M_x = M_o.
The book stresses that this combination is commutative: the order never matters. The same logic gives M_o \cdot M_x = M_x \cdot M_o = M_y, and M_o \cdot M_y = M_y \cdot M_o = M_x.
The table below collects the three rules and the sign each one changes. It is the fastest way to stop confusing them.
| Operation | Rule | Sign that changes |
|---|---|---|
| M_x (x-axis) | M_x(x, y) = (x, -y) | ordinate only |
| M_y (y-axis) | M_y(x, y) = (-x, y) | abscissa only |
| M_o (origin) | M_o(x, y) = (-x, -y) | both |
Common mistake: applying the x-axis rule to a y-axis question. Read the line named in the question first. If the question says origin, both signs flip.
This is the exam-pattern spot in the Class 10 ICSE syllabus. Board questions often give a combined reflection and ask for the single transformation that equals it. Two reflections in the axes, in either order, combine to M_o. Pair M_o with one axis and you get the other axis.
12.8 Invariant Point
An invariant point is a point that stays exactly where it is under a transformation. The book calls it a point that remains unaltered. For example, A(5, 0) reflected in the x-axis gives (5, 0) again, so A is invariant under M_x.
The pattern is simple: any point lying on the mirror line is its own image. Points on the x-axis are invariant under M_x. Points on the y-axis are invariant under M_y. The origin is invariant under all three: M_x, M_y and M_o.
This idea connects directly to the graph-paper questions in Exercise 12(B). Those questions hand you two invariant points and ask you to name the mirror line. If (-5, 0) and (4, 0) are both invariant, they lie on the x-axis, so the line is y = 0. If (0, -6) and (0, 5) are invariant, they lie on the y-axis, so the line is x = 0.
12.9 Using Graph Paper
Reflection on graph paper extends the rules to lines other than the axes. The line x = a is vertical, parallel to the y-axis and a units from it. The line y = a is horizontal, parallel to the x-axis and a units from it. These lines act as mirror lines too.
The method has three steps. Mark the given point. Draw a perpendicular from the point to the mirror line and continue it past the line. Mark the image at the same distance on the other side.
Worked pattern: reflect (1, 2) in the line x = 3. The point sits 2 units left of x = 3. Its image sits 2 units right of x = 3, at (5, 2). The ordinate 2 never changes because the mirror line is vertical.
For a horizontal line, the abscissa stays fixed. Reflect (4, 1) in the line y = -3. The point sits 4 units above the line, so the image sits 4 units below, at (4, -7).
This is where board questions concentrate in this ICSE chapter. Past papers ask you to plot a triangle, reflect it in a line, name the resulting figure — often a parallelogram, a trapezium or an isosceles trapezium — and find its area. Practise the three-step method until plotting is automatic, then the naming and area questions become routine. For the other subjects alongside this one, the Class 10 ICSE books list has them all.
The reflection questions in board papers follow the syllabus published on the official CISCE website.
What to practise from Reflection Class 10 ICSE
- EXERCISE 12(A) — 19 questions. It drills the foundations: the sign-change rules for M_x, M_y and M_o, invariant points, and combined reflections written as single transformations.
- EXERCISE 12(B) — 17 questions. It moves to graph paper: reflecting points and figures in lines like x = 2 and y = 3, naming the resulting figures, and past-board questions that ask for areas.
Work through 12(A) until the sign rules are instant. Then use 12(B) to practise the drawing method, because the figure-naming and area questions need accurate plots.
Reflection Class 10 ICSE frequently asked questions
What is an invariant point in reflection class 10 ICSE?
An invariant point is one that stays unchanged under a transformation. Any point on the x-axis is invariant under M_x, any point on the y-axis under M_y, and the origin under all three. Such a point is its own image.
What is the difference between reflection in the x-axis and reflection in the y-axis?
Reflection in the x-axis flips only the ordinate, giving (x, -y). Reflection in the y-axis flips only the abscissa, giving (-x, y). Remember the mnemonic: the axis you cross is the one you keep.
How do I find the image of a point when it is reflected in the origin?
Change the signs of both co-ordinates. For a point (x, y), the image is (-x, -y). So (2, -3) reflects to (-2, 3).
Does reflection in the x-axis followed by reflection in the y-axis equal reflection in the origin?
Yes. Applying M_x then M_y to (x, y) gives (-x, -y), which is exactly M_o. The combination is commutative, so the order does not matter.
How do I reflect a point in the line x = 2 on graph paper?
Draw the vertical line x = 2. Measure the point’s horizontal distance from it, then mark the image the same distance on the other side. A point at (-1, 3) is 3 units left, so its image is at (5, 3), 3 units right.
Reference: ICSE Class 10 Mathematics textbook material, chapter 12: Reflection — CISCE-aligned.
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