Reflection | ICSE Class 10 Maths Notes

This note covers coordinates, mirror images in the coordinate axes, reflection in lines parallel to the axes, reflection in the origin, invariant points, graphical construction and checks on reflected coordinates.

How do coordinates describe the position of a point?

The Cartesian plane is a flat surface equipped with two perpendicular coordinate axes. Perpendicular lines meet at a right angle. The horizontal axis is the x-axis; the vertical axis is the y-axis. Their intersection is the origin, denoted by O, with coordinates (0, 0).

An ordered pair gives two numbers in a specified order. For a point P(x, y), P names the point, x is its horizontal coordinate and y is its vertical coordinate. The horizontal coordinate is also called the abscissa; the vertical coordinate is the ordinate.

Coordinates include direction. A positive x-coordinate places a point to the right of the y-axis, and a negative one places it to the left. A positive y-coordinate places it above the x-axis, and a negative one places it below.

What do zero coordinates tell us?

A point on the x-axis has y-coordinate zero, so its coordinates have the form (x, 0). A point on the y-axis has x-coordinate zero, so its coordinates have the form (0, y). The origin satisfies both conditions.

Consequently, y = 0 is the equation of the x-axis and x = 0 is the equation of the y-axis. An equation describes a line here by stating the coordinate condition satisfied by every point on it.

How do coordinate signs locate a point?

The axes divide the plane into four regions called quadrants. The Roman numerals I, II, III and IV name these regions. Points on an axis are not inside a quadrant. Keep this distinction when a reflected point has a zero coordinate.

The order of the coordinates matters. Read the horizontal coordinate first and the vertical coordinate second. A reflection changes a position according to a geometrical rule; it does not give permission to exchange the entries of an ordered pair.

What does reflection in a line mean?

Definition: Reflection in a line maps a point to its mirror image. For a point off the line, the line is the perpendicular bisector of the segment joining the point and its image. Points on the line remain fixed.

A line segment is the straight portion between two endpoints. Its midpoint divides it into two equal lengths. A perpendicular bisector passes through that midpoint at a right angle to the segment. These ideas specify where a reflected point must lie.

Write P′ for the image of P; read the prime mark as “prime”. If P has coordinates (x, y), write its image as P′(x′, y′), where x′ and y′ are the image's coordinates. The mark names the image, not a numerical operation.

Property: Equal perpendicular distances

For a point off the mirror line, the original point and image lie on opposite sides of it at equal perpendicular distances. The segment joining them crosses the mirror line at its midpoint. Merely choosing a point somewhere on the other side does not establish reflection.

Distance is a non-negative length even when a coordinate is negative. The absolute value of a number is its distance from zero, written with vertical bars. Thus |x| means the absolute value of x and gives the distance of P(x, y) from the y-axis.

How can the image be constructed?

1. Identify the specified mirror line and plot the original point using its ordered coordinates.
2. If the point lies on the mirror line, keep its position unchanged and label the image there.
3. Otherwise, draw a perpendicular from the point to the mirror line and extend it across the line.
4. Mark the image at the same distance on the opposite side, then read its coordinates.

Coincident points occupy the same position. This happens when an original point lies on its mirror line. For such a point, the original and image coincide; there is no non-zero joining segment to bisect.

How is a point reflected in the x-axis?

The x-axis is the horizontal mirror line y = 0. Reflection across it moves a point vertically, so the horizontal coordinate stays unchanged. The vertical coordinate has the opposite sign because the image lies equally far above or below the axis.

Result: Reflection in y = 0

For P(x, y) with image P′(x′, y′), the coordinate rule is x′ = x and y′ = −y. In ordered-pair form, P(x, y) → P′(x, −y). The arrow means “maps to”, and the minus sign gives the opposite of the coordinate.

Changing a sign is not the same as making a number negative. When y is negative, −y is positive. When y is zero, −y is also zero. This last case explains why a point already on the x-axis does not move.

Worked example 1. Points M(−5, −2) and P(−5, 2) are given. Determine whether they are reflections of each other in the x-axis.

Answer: Both have x-coordinate −5. Their y-coordinates are opposites: −(−2) = 2. Each lies 2 units from y = 0, on opposite sides. Therefore P is the reflection of M in the x-axis.

Here M and P are the supplied names of the two points. Equal x-coordinates mean they are vertically aligned. The equality of their perpendicular distances completes the geometrical check; matching just one feature would not be enough.

Worked example 2. Find the reflection of B(4.5, 0) in the x-axis.

Answer: Keep the x-coordinate 4.5 and replace the y-coordinate by −0 = 0. The image B′ is (4.5, 0), the same position as B. The point lies on the mirror line.

For points off both axes, reflection in the x-axis exchanges quadrants I and IV, and exchanges quadrants II and III. This follows from keeping the sign of x and reversing the sign of y. Use it as a check after finding the coordinates.

Note: Reflection in the x-axis changes the y-coordinate. The name of the axis tells you the mirror line, not the coordinate whose sign must change.

How is a point reflected in the y-axis?

The y-axis is the vertical mirror line x = 0. Reflection across it moves a point horizontally. The image has the same height as the original point, so the y-coordinate stays unchanged. Equal distances on opposite sides require opposite x-coordinates.

Result: Reflection in x = 0

For P(x, y) and its image P′(x′, y′), use x′ = −x and y′ = y. Thus P(x, y) → P′(−x, y). A point whose x-coordinate is zero stays fixed because reversing the sign of zero leaves it unchanged.

Worked example 3. Reflect the triangle with vertices A(3, 4), D(7, 1) and M(9, 6) in the y-axis. A vertex is a corner of a figure.

Answer: Reverse each x-coordinate and retain each y-coordinate. The image vertices are A′(−3, 4), D′(−7, 1) and M′(−9, 6). Join the corresponding image vertices to obtain the reflected triangle.

The diagram shows A(3, 4), D(7, 1) and M(9, 6) to the right of the y-axis, and A′(−3, 4), D′(−7, 1) and M′(−9, 6) to its left. Each original vertex and its image share the same height.

See Fig. 1.9 in your NCERT textbook

Corresponding vertices are original vertices paired with their images. The reflected triangle preserves the lengths of corresponding sides. Applying one reflection rule to all its vertices keeps the pairings consistent and produces its mirror image.

Worked example 4. Find the image of H(0, 4) in the y-axis.

Answer: The image H′ has coordinates (−0, 4) = (0, 4). Its x-coordinate remains zero, so H′ coincides with H. Its height is unchanged, just as for every reflection in the y-axis.

For points off both axes, quadrants I and II exchange, as do quadrants III and IV. The y-coordinate retains its sign while the x-coordinate reverses its sign. A point on the y-axis remains there instead of moving into a quadrant.

To check a proposed answer, first compare heights and then compare distances to x = 0. Both checks are necessary. A point at the correct height but at a different horizontal distance is not the required image.

How does reflection in the origin differ from reflection in an axis?

Reflection in the origin is reflection in a point. For a point other than O, its image lies on the same straight line through O, on the opposite side, at an equal distance. In this case O is the midpoint of the joining segment.

The origin is a point, not a mirror line. Reflection in an axis reverses one coordinate. Reflection in the origin reverses both coordinates, because both horizontal and vertical displacements from O must be reversed. A displacement records a change in position, including its direction.

Result: Both coordinates change sign

For P(x, y) and image P′(x′, y′), the rule is x′ = −x and y′ = −y. Therefore P(x, y) → P′(−x, −y). Zero coordinates remain zero, while every non-zero coordinate changes to its opposite.

The midpoint formula provides a check. For endpoints P(x, y) and P′(x′, y′), the midpoint has coordinates ((x + x′)/2, (y + y′)/2). The slash means division. Each coordinate is the average of the corresponding endpoint coordinates.

Worked example 5. Let P(a, b) and Q(−a, −b) be points, where a and b are real numbers, meaning numbers on the number line. Show that they are reflections of each other in the origin.

Answer: The midpoint is ((a + (−a))/2, (b + (−b))/2) = (0, 0). Both coordinates of Q are the opposites of those of P. Thus Q is the reflection of P in O, including the coincident case a = b = 0.

For points off both axes, reflection in O exchanges quadrants I and III, and exchanges quadrants II and IV. Unlike reflection in either axis, both coordinate signs reverse. A point on an axis remains on that same axis, on the opposite side of O unless it is O itself.

The same final coordinates result from reflection in the x-axis followed by reflection in the y-axis: (x, y) becomes (x, −y), then (−x, −y). Reversing the order gives the same result. This describes the particular pair of coordinate axes.

Note: Equal distances from O alone do not prove that two points are images in O. Their joining segment must also have O as its midpoint, unless both points coincide with O.

How is a point reflected in the vertical line x = a?

In this section, a is a fixed real number locating the mirror line. The equation x = a describes all points whose horizontal coordinate is a, whatever their vertical coordinate. It is a vertical line, parallel to the y-axis when a is non-zero. Parallel lines lie in the same plane and do not meet however far they are extended.

The image of P(x, y) lies horizontally across this mirror line. Its y-coordinate stays y. The x-coordinate is reflected about a rather than about zero, so simply changing the sign of x does not give the general answer.

How is the coordinate rule derived?

Let P′(x′, y′) be the image, using x′ and y′ for its coordinates as before. The perpendicular joining segment is horizontal, and its midpoint has x-coordinate a. Hence the average of x and x′ must equal a.

1. Keep the vertical coordinate unchanged: y′ = y, because the original and image lie at the same height.
2. Use the midpoint condition on the horizontal coordinates: (x + x′)/2 = a.
3. Multiply by 2 to obtain x + x′ = 2a, where 2a means twice a.
4. Subtract x to obtain x′ = 2a − x, and write the image as P′(2a − x, y).

Thus the rule is P(x, y) → P′(2a − x, y). It works on either side of the line. If x equals a, substitution returns a as the image's horizontal coordinate, so a point on the mirror line remains fixed.

Worked example 6. A point P(x, y) is reflected in x = a, where x and y are its coordinates and a is a fixed real number. Find its image and check the midpoint.

Answer: The image is P′(2a − x, y). The horizontal midpoint coordinate is (x + 2a − x)/2 = a, and the vertical midpoint coordinate is (y + y)/2 = y. The midpoint is therefore (a, y), on the stated mirror line.

Its signed horizontal displacement from the line is reversed: x′ − a = a − x, the opposite of x − a. Consequently, the perpendicular distances |x′ − a| and |x − a| are equal. Absolute value removes the directional sign when calculating distance.

Setting a = 0 gives x′ = −x, recovering reflection in the y-axis. This checks both the general formula and the identification of the correct coordinate axis.

How is a point reflected in the horizontal line y = a?

Here a is a fixed real number giving the mirror line's height. The equation y = a describes all points with vertical coordinate a. It is a horizontal line, parallel to the x-axis when a is non-zero.

The original point and image are vertically aligned, so the x-coordinate is unchanged. Their vertical coordinates lie equally far from a on opposite sides. As with a vertical mirror line, the relevant coordinate is reflected about a, rather than necessarily about zero.

How is the horizontal-line rule obtained?

For P(x, y) and image P′(x′, y′), the midpoint of the vertical joining segment lies on y = a. The average of y and y′ is therefore a. Solving this condition gives the required image coordinate.

1. Retain the horizontal coordinate: x′ = x. The original and image lie on the same vertical line.
2. Set the vertical midpoint coordinate equal to the mirror height: (y + y′)/2 = a.
3. Multiply by 2 to obtain y + y′ = 2a, then subtract y to obtain y′ = 2a − y.
4. Write both coordinates in their original order: P′(x, 2a − y).

Worked example 7. Reflect P(x, y) in y = a, where a is a fixed real number. Verify that the perpendicular distances from the mirror line agree.

Answer: P′ is (x, 2a − y). The original distance is |y − a|. The image distance is |(2a − y) − a| = |a − y| = |y − a|. The signed vertical displacements are opposites, and both points have the same x-coordinate.

When y = a, the rule gives y′ = 2a − a = a, so the point remains fixed. When a = 0, it gives y′ = −y, which is the rule for reflection in the x-axis.

Draw the axes and the line y = a. Place P(x, y) off that line and draw the vertical perpendicular through it. Mark P′(x, 2a − y) equally far across the line and label their midpoint (x, a).

Compare the two parallel-line rules carefully: a vertical mirror changes the horizontal coordinate; a horizontal mirror changes the vertical coordinate. In both cases, double the fixed coordinate of the mirror and subtract the corresponding original coordinate.

Which points are invariant under a reflection?

Definition: An invariant point is a point whose image under the specified transformation is itself. A transformation is a rule that maps original points to image points.

The words under the specified transformation matter. A point may stay fixed in one reflection and move in another. Invariance concerns the whole point: both image coordinates must equal the corresponding original coordinates.

Property: A mirror line consists of invariant points

Every point on a mirror line stays fixed under reflection in that line. A point off it moves to the opposite side, so it is not invariant. Thus the invariant points are exactly the points on the mirror line.

This gives a direct algebraic test. For reflection in x = a, set the image's horizontal coordinate 2a − x equal to x. Solving gives x = a. The y-coordinate is unrestricted because reflection already leaves it unchanged.

For reflection in y = a, solve 2a − y = y to obtain y = a. There is no restriction on x. “Unrestricted” means that the coordinate can have any real value while the other coordinate satisfies the stated condition.

Worked example 8. Find all points P(x, y) invariant under reflection in the origin.

Answer: The image is (−x, −y). Equality with (x, y) requires −x = x and −y = y. These give 2x = 0 and 2y = 0, so x = 0 and y = 0. Only O(0, 0) is invariant.

A point invariant under reflection in both coordinate axes must satisfy both axis conditions. It must have x = 0 and y = 0, so it is also O. Do not replace “both” with “either”: the latter permits points elsewhere on either axis.

How can reflected coordinates be plotted and checked?

Begin by identifying whether the reflection is in a line or in the origin. Then write the applicable rule before doing any substitution. This separates the geometrical decision from the arithmetic and reduces confusion between a mirror's name and the coordinate that changes.

Which rule matches the specified reflection?

In the table, a is the fixed real coordinate locating the line. The centre of the origin reflection is O, the point about which both coordinates reverse. A centre and a mirror line specify different geometrical conditions.

1. Draw perpendicular axes, mark the origin and use equal coordinate units on both axes.
2. Draw and label the mirror line, or identify O when reflecting in the origin.
3. Plot the original point, calculate its image and plot the image using the same scale.
4. Check alignment, equal perpendicular distances and the appropriate midpoint condition. Label the original and image distinctly.

How can the mirror be recovered from a point and its image?

For distinct points P(x, y) and P′(x′, y′) with y′ = y, reflection in a vertical line requires a = (x + x′)/2. For distinct points with x′ = x, a horizontal mirror requires a = (y + y′)/2.

The condition that the points are distinct is essential. A point that coincides with its image does not by itself determine a unique mirror line. Instead, any proposed mirror line must pass through that fixed point.

Reflecting an image again in the same mirror line returns the original point. For x = a, the restored horizontal coordinate is 2a − (2a − x) = x. For y = a, the corresponding calculation is 2a − (2a − y) = y.

This reverse check also works for reflection in O because changing both signs twice restores both original coordinates. Use the complete ordered pair when checking; agreement of just one coordinate cannot establish that the original point has been recovered.

Glossary

• Cartesian plane — A plane equipped with perpendicular coordinate axes for locating points using ordered pairs of numbers.
• Coordinate axes — The horizontal x-axis and vertical y-axis used together to specify positions in the coordinate plane.
• Origin — The intersection of the coordinate axes, with both coordinates equal to zero.
• Ordered pair — Two numbers written in a specified order, giving the horizontal coordinate followed by the vertical coordinate.
• Abscissa — The x-coordinate of a point, specifying its horizontal position relative to the y-axis.
• Ordinate — The y-coordinate of a point, specifying its vertical position relative to the x-axis.
• Quadrant — One of the four regions into which the coordinate axes divide the plane.
• Reflection in a line — A transformation giving mirror images across a specified line while fixing points on that line.
• Image — The point obtained from an original point by applying the specified transformation.
• Midpoint — The point on a line segment that divides it into two equal lengths.
• Perpendicular bisector — A line through a segment's midpoint that meets the segment at a right angle.
• Invariant point — A point whose image under the specified transformation coincides with the original point.
• Absolute value — The non-negative distance of a number from zero on the number line.
• Reflection in the origin — A point reflection in which both coordinates reverse sign and the origin remains fixed.

Common errors and misconceptions

• Misconception: Reflection in the x-axis changes the x-coordinate's sign. Correct: It keeps x unchanged and reverses y; the movement is perpendicular to the horizontal mirror line.
• Misconception: The line x = 0 is the x-axis. Correct: It is the y-axis, because every point on the vertical axis has horizontal coordinate zero.
• Misconception: Reflection in x = a gives the horizontal coordinate a − x. Correct: It gives 2a − x, because a is the average of the original and image coordinates.
• Misconception: The origin is a mirror line. Correct: It is a point. Reflection in it reverses both coordinates and places O at the midpoint of a non-zero joining segment.
• Misconception: A point is invariant if one coordinate stays unchanged. Correct: Both image coordinates must match the original coordinates for the whole point to remain fixed.
• Misconception: Changing a coordinate's sign makes it negative. Correct: A negative coordinate becomes positive when its sign is reversed, and zero remains zero.
• Misconception: Points equally far from a mirror line must be images. Correct: Distinct reflected points must also lie on the same perpendicular, on opposite sides of that line.

Exam-style questions with model answers

Q1. Find the reflection of A(3, 4) in the y-axis and explain which coordinate is unchanged. [2 marks]

1. The y-axis reflection rule changes the sign of the horizontal coordinate, giving the image A′(−3, 4).
2. The y-coordinate remains 4 because reflection in a vertical axis keeps the point at the same height.

Q2. Given M(−5, −2) and P(−5, 2), identify the coordinate axis in which they are reflections and justify the answer. [3 marks]

1. The mirror is the x-axis, whose equation is y = 0. The two points share x-coordinate −5, so the segment joining them is vertical and perpendicular to this axis.
2. Their y-coordinates are −2 and 2. They lie on opposite sides of the axis, each at a perpendicular distance of 2 units.
3. The midpoint is (−5, 0), on the x-axis. The axis therefore bisects the joining segment perpendicularly, establishing the required reflection.

Q3. B(4.5, 0) is reflected in the x-axis, and H(0, 4) is reflected in the y-axis. Give both images and state, with reasons, whether each point is invariant. [4 marks]

1. For B, reflection in the x-axis retains 4.5 as the horizontal coordinate and changes the vertical coordinate from 0 to −0 = 0. Thus B′ is (4.5, 0).
2. B is invariant because its image has the same coordinates as B. Its zero ordinate places it on the specified mirror line.
3. For H, reflection in the y-axis changes 0 to −0 = 0 while retaining the vertical coordinate 4. Thus H′ is (0, 4).
4. H is invariant because its image coincides with H. Its zero abscissa places it on the y-axis, its specified mirror line.

Q4. Triangle ADM has vertices A(3, 4), D(7, 1) and M(9, 6). Reflect it in the y-axis: give each image vertex, check A and its image geometrically, and state what happens to corresponding side lengths. [5 marks]

1. A(3, 4) maps to A′(−3, 4). Reflection in the y-axis reverses the horizontal coordinate while preserving the vertical coordinate.
2. D(7, 1) maps to D′(−7, 1). Its height remains 1, and its horizontal coordinate becomes the opposite of 7.
3. M(9, 6) maps to M′(−9, 6). Join A′, D′ and M′ to form the image triangle with the same corresponding vertex pairings.
4. A and A′ share height 4 and are each 3 units from the y-axis on opposite sides. Their midpoint (0, 4) lies on the axis, and their joining segment is horizontal.
5. Corresponding side lengths remain equal under reflection. The reflected triangle has the same side lengths as the original triangle, with each side paired with the side joining its image endpoints.

Q5. P(x, y), with real coordinates x and y, is reflected in the vertical line x = a, where a is a fixed real number. Find the image, derive its changed coordinate, and obtain the condition for P to be invariant. [5 marks]

1. Write the image as P′(x′, y′), where x′ and y′ are its coordinates. Since the mirror is vertical, original and image have the same height, giving y′ = y.
2. The mirror bisects the horizontal joining segment when the points are distinct. Hence its fixed coordinate a equals their average horizontal coordinate: (x + x′)/2 = a. This also holds for a fixed point.
3. Multiply by 2 and subtract x. This gives x + x′ = 2a, followed by x′ = 2a − x.
4. Combining the changed and unchanged coordinates gives the complete image P′(2a − x, y), in horizontal-then-vertical coordinate order.
5. For invariance, require 2a − x = x. Therefore 2x = 2a and x = a; y is unrestricted. These are precisely the points on the mirror line.

Q6. P(x, y), with real coordinates x and y, is reflected in the horizontal line y = a, where a is fixed and real. Find its image, verify equal perpendicular distances and state the condition for invariance. [3 marks]

1. The image is P′(x, 2a − y). A horizontal mirror leaves the horizontal coordinate unchanged, and the original and image vertical coordinates have average a.
2. The original perpendicular distance is |y − a|. The image distance is |(2a − y) − a| = |a − y| = |y − a|, so the distances agree.
3. For invariance, require 2a − y = y, which gives y = a. The horizontal coordinate x can be any real number.

Q7. P(a, b) and Q(−a, −b) have real coordinates determined by a and b. Show that Q is the reflection of P in the origin, and determine when the two points coincide. [3 marks]

1. Reflection in the origin reverses both coordinates. Applied to P(a, b), it gives (−a, −b), which are the stated coordinates of Q.
2. The midpoint check gives ((a + (−a))/2, (b + (−b))/2) = (0, 0). Thus the origin is the midpoint, as required for this reflection.
3. Coincidence requires a = −a and b = −b, giving a = 0 and b = 0. Both points then coincide with O(0, 0).

Key takeaways

• Coordinates are ordered: read the horizontal coordinate first and the vertical coordinate second when locating an original point or image.
• Reflection in a line places a point off the line equally far across it along the same perpendicular.
• Reflection in the x-axis keeps the horizontal coordinate unchanged and reverses the sign of the vertical coordinate.
• Reflection in the y-axis keeps the vertical coordinate unchanged and reverses the sign of the horizontal coordinate.
• Reflection in the origin reverses both coordinates, making the origin the midpoint of each non-zero joining segment.
• For reflection in x = a, the image is (2a − x, y); for y = a, it is (x, 2a − y).
• Invariant points of a line reflection lie on its mirror line; the origin reflection has just one invariant point.
• Check the complete ordered pair, the midpoint condition and perpendicular distances; a correct-looking sketch alone is not a coordinate calculation.

Test yourself

Which coordinate axis has equation x = 0?

The y-axis has equation x = 0 because all its points have horizontal coordinate zero.

What is the image of P(x, y) in the x-axis?

The image is P′(x, −y): keep the horizontal coordinate and reverse the vertical coordinate's sign.

What is the image of P(x, y) in the y-axis?

The image is P′(−x, y): reverse the horizontal coordinate's sign and keep the vertical coordinate.

Where is the midpoint of a point and its reflection in the origin?

The midpoint is O(0, 0), since each coordinate and its opposite have average zero.

For a fixed real number a, which points are invariant under reflection in x = a?

Every point (a, y) is invariant, with y any real number, because it lies on the mirror line.

For a fixed real number a, what is the reflection of P(x, y) in y = a?

The image is P′(x, 2a − y), with unchanged horizontal coordinate and average vertical coordinate a.

How many points are invariant under reflection in the origin, and which are they?

There is one invariant point, O(0, 0), because both coordinates must equal their own opposites.

What happens when a point is reflected twice in the same mirror line?

The second reflection returns the original point, reversing the first reflection's change in position.