Chapter 1

Orienting Yourself: The Use of Coordinates

How to describe the exact position of anything on a flat surface using just two numbers. By the end, you will be able to find any point, name any point, and measure the straight distance between any two points.

Why words like "near the window" are not exact enough, and how numbers fix that
How two number lines, crossing at zero, make a map of a whole flat surface
How to read and write a point's "address", like (3, −2)
The four quadrants and the sign pattern in each one
How to find the distance between two points using the Baudhāyana-Pythagoras theorem
What happens to points when they are reflected in an axis, like in a mirror

Class 9, MathematicsOpen the original PDF

1.1Why do we need coordinates?

Imagine you lose your pencil in the classroom and ask a friend where it is. Your friend says, "It is somewhere near the back." That helps a little, but you still have to search. Now imagine your friend says, "Third row from the front, second desk from the left." You can walk straight to it.

The second answer works because it uses two numbers, each counted from a fixed starting place. One number tells you how far to go in one direction (rows). The other tells you how far to go in the other direction (desks). Together, they point to exactly one spot.

You already use this idea

A cinema ticket says "Row F, Seat 12". A chessboard square is called "e4". A classroom seating chart says "row 3, bench 2". In every case, two pieces of information pin down one exact place. Coordinates are simply the mathematical way of doing the same thing.

Coordinate system

A fixed framework, like the grid lines on graph paper or on a map, that lets us describe the exact location of any point using numbers.

The numbers that describe a point's location are called its coordinates. You can think of them as the point's address. Just like your home address lets a delivery person find your house, a point's coordinates let anyone find that point.

1.1A short journey through history

This idea of using a grid to locate things is very old, and India played a big part in it. You do not need to memorise these dates. Read this as a story of how one good idea grew over thousands of years.

About 4,500 years ago: the Sindhu-Sarasvatī CivilisationCities were planned with streets running exactly North-South and East-West, placed about 10 metres apart. A merchant could find a shop by counting how many streets North or South, and how many East or West, from the city centre. That is a coordinate system used in real life.
Around 800 BCE: BaudhāyanaHe used East-West and North-South lines to make careful geometric constructions. His work includes the rule we now call the Baudhāyana-Pythagoras theorem, which we will use later in this chapter to measure distances.
By the 4th century BCE: Ujjayinī as the "zero line"Early Indian astronomy books (the Siddhāntas) treated the city of Ujjayinī (Ujjain) as the starting line from which the positions of other places on Earth were measured.
Around 499 CE: ĀryabhaṭaHe replaced older Greek methods with the "sine", which made it much easier to calculate the coordinates of a star or a city. He even mapped the sky using coordinates.
Around 628 CE: BrahmaguptaHe treated zero and negative numbers as proper numbers you can calculate with. This matters a lot for us: in a coordinate system, the starting point is zero, and the directions to the left and downward use negative numbers. Without his work, the full coordinate plane in this chapter would not be possible.
Around 1000 CE: Al-BīrūnīThis scholar travelled to India, studied the Siddhāntas, and used Indian methods to calculate the coordinates of cities across Asia. He also improved the astrolabe, a handheld tool sailors used to find their position from the stars.
Around 1100 CE: Omar KhayyamHe was the first to solve algebra problems by turning them into shapes on a coordinate plane.
1636 and 1637 CE: Fermat and DescartesThese two French mathematicians showed that any point on a flat surface can be described by just two numbers: its distances from two lines that cross at a right angle. This joined algebra (equations) and geometry (shapes) together. The coordinate plane is often called the Cartesian plane after Descartes.

The one thing to take from history

Two numbers, measured from a fixed starting point along two directions at right angles, are enough to locate any point on a flat surface. Everything else in this chapter builds on that single idea.

1.2Meet Reiaan: a map you can touch

Reiaan's family has just moved to a new city. Reiaan is visually impaired, which means he cannot see, so a new home and a new school can feel confusing at first. His older sister Shalini, who has just finished Class 9, decides to use what she learnt about coordinates to help him.

She takes a board with a grid on it and builds a model of his room. She pushes pins into the board to mark important points, like the corners of the room, the door, and the furniture. Then she stretches thick wool between the pins, so Reiaan can run his fingers along the walls and objects and feel where everything is.

Shalini's touch map: pins mark the corners, and wool threads trace the walls and furniture.

Scale

Shalini used a scale of 1 cm : 1 foot. This means every 1 centimetre on her board stands for 1 foot in the real room. So if two pins are 4 cm apart on the board, the real objects are 4 feet apart in the room.

The board shows only the floor of the room, as if you were a bird looking straight down from the ceiling. That brings up an interesting question from the book.

Why can the windows not be marked on this floor map?Tap to reveal

A floor map is flat. It only shows two directions: left and right, and front and back. Windows are on the walls, some height above the floor. Height is a third direction (up and down), and a flat floor map has no way to show it. At most, you could mark where on the wall a window is, but not how high it is.

Shalini's board is really a coordinate system. Let us now learn exactly how such a system is built.

1.3From a number line to a flat plane

You already know the number line from earlier classes. It is a straight line with zero in the middle. Numbers to the right of zero are positive. Numbers to the left of zero are negative.

A number line. The point 3 is three steps to the right of zero. The point −2 is two steps to the left of zero. One number is enough, because you can only move left or right.

A number line is one-dimensional (1-D). "One dimension" means there is only one direction you can move in: along the line. So one number is enough to describe any position on it.

But a floor, a sheet of paper, or a map is flat and wide. You can move left and right, but also forward and back. That is two dimensions (2-D). If someone says "the pen is 3 steps away", you still do not know which way to walk. You need a second number.

The clever trick

Take two number lines. Lay one flat (horizontal). Stand the other one straight up (vertical). Make them cross exactly at their zeros, at a right angle. Now you can describe any spot on the flat surface: "go this far along the flat line, then this far along the upright line".

The parts of the coordinate plane

x-axis

The horizontal number line (the one lying flat, going left and right).

y-axis

The vertical number line (the one standing up, going up and down).

Origin (O)

The point where the two axes cross. It is the starting point for all measurements. Its coordinates are (0, 0), because you have not moved at all in either direction.

The word axes (said "ak-seez") is simply the plural of axis. "The coordinate axes" means the x-axis and the y-axis together.

Which way is positive, and which way is negative?

Both axes are marked in equal steps (units) starting from O. The rule for signs is the same as on a normal number line, and it is worth learning by heart:

Direction from O	Sign	Which axis
To the right	Positive (+)	x-axis
To the left	Negative (−)	x-axis
Upward	Positive (+)	y-axis
Downward	Negative (−)	y-axis

Easy way to remember

Think of a lift and a corridor. Going up in a lift feels like "more" (positive). Going down to the basement feels like "less" (negative, like floor −1). In the corridor, walking right is positive, and walking left is negative, just like on the number line you already know.

1.3Writing a point's address: (x, y)

Every point on the plane gets a pair of numbers written inside brackets, separated by a comma, like this: (x, y).

The first number, x, is called the x-coordinate. It tells you how far to go left or right from O.
The second number, y, is called the y-coordinate. It tells you how far to go up or down.

Order matters: across first, then up or down

Always read the pair in the same order. First move across (x). Then move up or down (y). A handy memory line: "You walk into the building before you take the lift." Also, x comes before y in the alphabet, and x comes first in the brackets.

How to plot a point, step by step

Example: plot P (4, 3)

Put your finger on the origin O (0, 0).
Look at the first number, 4. It is positive, so move 4 steps to the right along the x-axis.
Look at the second number, 3. It is positive, so from where you are now, move 3 steps up.
Mark the point there and label it P (4, 3).

Example: plot Q (−5, −2)

Start at O.
The first number is −5. Negative means left, so move 5 steps to the left.
The second number is −2. Negative means down, so move 2 steps down.
Mark the point and label it Q (−5, −2).

What the coordinates really measure

The book describes coordinates like this: x is the perpendicular distance of the point from the y-axis, and y is the perpendicular distance of the point from the x-axis. "Perpendicular" means "at a right angle", which is the straightest, shortest way.

In simple words: the x-coordinate tells you how far the point is from the upright line (the y-axis), and which side it is on. The y-coordinate tells you how far the point is from the flat line (the x-axis), and whether it is above or below.

Common mix-up

Students often think "the x-coordinate is the distance from the x-axis". It is the opposite. The x-coordinate is measured along the x-axis, so it tells you how far you are from the y-axis. For example, the point (4, 3) is 4 steps away from the y-axis and 3 steps away from the x-axis.

One more small thing: when marking points on a graph, people often drop the "=" sign. So instead of writing P = (4, 3), you can write P (4, 3). Both mean the same.

1.3Points that sit on the axes

Some points lie exactly on one of the axes. These are easy to spot once you notice a pattern. Look at the diagram below, which is a redrawn version of Fig. 1.2 in the book.

Points on the axes. B and E are on the x-axis, so they have not moved up or down at all. H and G are on the y-axis, so they have not moved left or right at all.

B (4.5, 0): 4.5 steps to the right of O, and 0 steps up or down. So it sits on the x-axis.
E (−2.9, 0): 2.9 steps to the left of O, and 0 steps up or down. Also on the x-axis.
H (0, 4): 0 steps left or right, then 4 steps up. So it sits on the y-axis.
G (0, −4.5): 0 steps left or right, then 4.5 steps down. Also on the y-axis.

The pattern

Any point on the x-axis looks like (x, 0): its y-coordinate is always 0, because it has not gone up or down.

Any point on the y-axis looks like (0, y): its x-coordinate is always 0, because it has not gone left or right.

So for a point P (x, 0) on the x-axis: if x is positive, P is to the right of O; if x is negative, P is to the left. For a point P (0, y) on the y-axis: if y is positive, P is above O; if y is negative, P is below.

Notice also that coordinates do not have to be whole numbers. B is at 4.5 and E is at −2.9. Points can sit anywhere, including between the grid lines.

Ex 1.1Practice: Reiaan's room

The book's Exercise Sets 1.1 and 1.2 use a drawing of Reiaan's room (Figs. 1.3 and 1.5). The book prints that drawing as a picture, so this page redraws it. The coordinates come from the book's figure as reported in published solution guides, which we cross-checked against each other and against the numbers the book's own text gives.

Each small square is 1 foot. The corners of Reiaan's room are O, A, B and C. The bathroom is on the left, with corners O, F, R and P. Inside the room are a bed (corners starting at S1 and S2), a wardrobe (corners starting at W1 and W2) and a study table.

Reiaan's room, redrawn. Green: the room door D1R1, on the bottom wall. Red: the bathroom door B1B2, on the wall shared with the bathroom. Room: O (0, 0), A (12, 0), B (12, 10), C (0, 10). Bathroom: O (0, 0), F (0, 9), R (−6, 9), P (−6, 0). Shower: S (−6, 6), H (−3, 6), W (−2, 9), R (−6, 9). Bed: (1, 5) to (7, 8). Wardrobe: (3, 1) to (7, 2). Door ends: D1 (8, 0) and R1 (11.5, 0).

(i) How far is the door D1R1 from the left wall (the y-axis)? How far is it from the x-axis?Tap to reveal

The nearer end of the door is D1 (8, 0). Its x-coordinate is 8, and the x-coordinate is the distance from the y-axis. So the door starts 8 feet from the left wall.

Its y-coordinate is 0, so the door is 0 feet from the x-axis. In other words, the door lies right on the x-axis, which is the bottom wall OA of the room.

(ii) What are the coordinates of D1?Tap to reveal

D1 = (8, 0). It is 8 steps right of O and 0 steps up.

(iii) R1 is (11.5, 0). How wide is the door? Is it comfortable? Can a wheelchair get through?Tap to reveal

Both ends are on the x-axis, so the width is just the difference of the x-coordinates: 11.5 − 8 = 3.5 feet.

Most room doors are between about 2.5 and 3 feet wide, so 3.5 feet is a generous, comfortable width.

For a wheelchair, most accessibility guidelines ask for a clear opening of at least about 90 cm, which is close to 3 feet. This door is 3.5 feet, about 107 cm, so yes, a wheelchair user can enter easily.

(iv) The bathroom door runs from B1 (0, 1.5) to B2 (0, 4). Is it narrower or wider than the room door?Tap to reveal

Both ends are on the y-axis (x = 0), so the width is the difference of the y-coordinates: 4 − 1.5 = 2.5 feet.

The room door is 3.5 feet wide, so the bathroom door is narrower, by 1 foot. (A 2.5 foot door would also be too narrow for most wheelchairs.)

Think and reflect

Measure a few doors at home and at school. Are they wide enough for someone in a wheelchair? Many older buildings have doors that are too narrow, and a simple measurement like this is how people find out which doors need to be changed.

1.3The four quadrants

So far we looked at points that sit on the axes. Most points are not on either axis. They are somewhere in the open space in between.

The flat surface on which we draw the axes is called the Cartesian plane. It is also called the coordinate plane or the xy-plane. All three names mean the same thing.

The two axes cut this plane into four parts, like a cross cutting a sheet of paper into four pieces. Each part is called a quadrant ("quad" means four). They are numbered with Roman numerals I, II, III and IV.

Quadrant II(−, +)left and up

Quadrant I(+, +)right and up

Quadrant III(−, −)left and down

Quadrant IV(+, −)right and down

How the numbering goes

Start at the top right (Quadrant I) and go round anticlockwise: top left is II, bottom left is III, bottom right is IV. It goes the opposite way to a clock's hands, like tracing the letter C backwards from the top right.

The signs follow directly from the rules we learnt. In Quadrant I you go right (+) and up (+). In Quadrant II you go left (−) and up (+). In Quadrant III you go left (−) and down (−). In Quadrant IV you go right (+) and down (−).

Redrawn Fig. 1.4. Q (−5, 3) is 5 left and 3 up, so it is in Quadrant II. S (3, −5) is 3 right and 5 down, so it is in Quadrant IV.

Watch out

A point that lies on an axis is not in any quadrant. For example, (0, 4) sits on the y-axis, which is the border between Quadrants I and II. It belongs to neither. Quadrants are only the open spaces between the axes.

Think and reflect: answered

1. What is the x-coordinate of any point on the y-axis?Tap to reveal

It is always 0. To stay on the y-axis you must not move left or right at all.

2. Is there a similar rule for points on the x-axis?Tap to reveal

Yes. Every point on the x-axis has y-coordinate 0, because it has not moved up or down.

3. Can the point Q (y, x) ever be the same as P (x, y)?Tap to reveal

Only when x and y are the same number. For example, (4, 4) swapped is still (4, 4). But (2, 5) swapped is (5, 2), which is a different point: one is 2 right and 5 up, the other is 5 right and 2 up.

4. True or false: if x ≠ y then (x, y) ≠ (y, x), and (x, y) = (y, x) only when x = y.Tap to reveal

True. This is exactly what we found in question 3. Two points are the same only when both their x-coordinates match and both their y-coordinates match.

1.3Why the order inside the brackets matters

This is so important that it deserves its own picture. The pairs (2, 5) and (5, 2) use the same two numbers, but they land in very different places.

Same numbers, different points. For (2, 5) you walk 2 across then 5 up (solid path). For (5, 2) you walk 5 across then 2 up (dashed path).

That is why coordinates are called an ordered pair: the order is part of the meaning.

Ex 1.2Practice: planning Reiaan's house

This is the book's Exercise Set 1.2. It uses the same room drawing (the book's Fig. 1.5), so scroll up to the redrawn plan if you need to look at it again.

1. Three feet of the study table are at (8, 9), (11, 9) and (11, 7). Where is the fourth foot? Is it a good spot? What are the table's width and length? Can you tell its height?Tap to reveal

Fourth foot. A rectangular table has sides that run straight across and straight up. The points (8, 9) and (11, 9) share y = 9, so that is the back edge. The points (11, 9) and (11, 7) share x = 11, so that is the right edge. The missing corner must be straight below (8, 9) and level with (11, 7). That gives (8, 7).

Size. Across: 11 − 8 = 3 feet. Up and down: 9 − 7 = 2 feet. So the table is 3 feet long and 2 feet wide.

Good spot? Yes. Check it against everything else in the room. The bed runs from x = 1 to x = 7, and the table starts at x = 8, so they do not overlap. The wardrobe is down at y = 1 to 2, far below the table. The table stays inside the room (x up to 12, y up to 10). And it sits in the upper right part of the room, away from the door at the bottom, so the path into the room stays clear.

Height? No. A floor plan is 2-D. It shows length and width only. Height is a third direction that a flat plan cannot show.

2. The bathroom door is hinged at B1 and opens into the bedroom. Will it hit the wardrobe? What if the door is made wider?Tap to reveal

The door is 2.5 feet wide (from y = 1.5 to y = 4). When it swings open around the hinge B1 (0, 1.5), its free edge moves along a quarter circle of radius 2.5 feet. When fully open, the door lies flat along the line y = 1.5, reaching out to the point (2.5, 1.5).

The wardrobe's left side is the line x = 3, running from W1 (3, 1) up to (3, 2). The point of the wardrobe closest to the hinge is (3, 1.5), which is exactly 3 feet from B1. The door only reaches 2.5 feet. So the door will not hit the wardrobe, with half a foot to spare.

If the door were made 3 feet or wider, it would reach the wardrobe and hit it. Possible fixes: move the wardrobe a little to the right, hinge the door at B2 instead so it swings upward (away from the wardrobe, and still short of the bed, which starts at y = 5), or use a sliding door.

3. Reiaan's bathroom: corner coordinates, the shape of the shower SHWR, and space for a washbasin and toilet.Tap to reveal

(i) Corners: O (0, 0), F (0, 9), R (−6, 9), P (−6, 0). The bathroom is 6 feet across and 9 feet deep.

(ii) Shower SHWR: S (−6, 6), H (−3, 6), W (−2, 9), R (−6, 9).

SH is horizontal (both at y = 6) and 3 feet long.
RW is horizontal (both at y = 9) and 4 feet long.
SR is vertical (both at x = −6).
HW is slanted: it goes from (−3, 6) to (−2, 9).

So SH and RW are parallel but have different lengths, and the other two sides are not parallel. A four-sided shape with exactly one pair of parallel sides is a trapezium. Because side SR meets both parallel sides at right angles, it is a right trapezium.

(iii) One possible layout (yours can be different, as long as nothing overlaps and the door is not blocked):

Washbasin, 3 ft by 2 ft: (−6, 0), (−3, 0), (−3, 2), (−6, 2).
Toilet, 2 ft by 3 ft: (−2, 6), (0, 6), (0, 9), (−2, 9).

Check: the door is on x = 0 between y = 1.5 and y = 4. The toilet starts at y = 6 and the basin stops at x = −3, so neither blocks the entrance. The toilet also stays clear of the shower, whose slanted side HW never goes further right than x = −2.

4. The dining room is 18 ft long and 15 ft wide, with its length from P to A. Find its corners, then place a 5 ft by 3 ft table exactly in the middle.Tap to reveal

(i) Corners. P is (−6, 0) and A is (12, 0). The distance from P to A is 12 − (−6) = 12 + 6 = 18 feet. That matches the length. The room door D1R1 is on the bottom wall of the bedroom, so the dining room lies below the x-axis, and it goes 15 feet down. Its corners are:

P (−6, 0), A (12, 0), (12, −15), (−6, −15)

(ii) Centre. The middle of the room is halfway across and halfway down:

Halfway across: (−6 + 12) ÷ 2 = 6 ÷ 2 = 3
Halfway down: (0 + (−15)) ÷ 2 = −7.5

So the centre is (3, −7.5). The table is 5 feet long, so it goes 2.5 feet each side of x = 3, from x = 0.5 to x = 5.5. It is 3 feet wide, so it goes 1.5 feet each side of y = −7.5, from y = −9 to y = −6. The four feet are:

(0.5, −9), (5.5, −9), (5.5, −6), (0.5, −6)

The red dot is the centre (3, −7.5). The table is spread equally around it.

1.4Distance along a straight row or column

Before the general method, start with the easy case: two points on the same horizontal line or the same vertical line.

Same y-coordinate (a horizontal line)

Take A (−3, 2) and B (4, 2). Both have y = 2, so they are at the same height. To get from A to B you only walk across. How far? Count from −3 up to 4: that is 7 steps. As a calculation: 4 − (−3) = 4 + 3 = 7.

Same x-coordinate (a vertical line)

Take C (1, −4) and D (1, 5). Both have x = 1, so one is straight above the other. The distance is 5 − (−4) = 5 + 4 = 9.

The book's example: Reiaan's furniture

The book mentions the distances W1W2 and S1S2 in Reiaan's room. Look back at the room plan:

The wardrobe edge W1W2 goes from W1 (3, 1) to W2 (7, 1). Same y, so W1W2 = 7 − 3 = 4 feet. That is the length of the wardrobe.
The bed edge S1S2 goes from S1 (1, 5) to S2 (7, 5). Same y, so S1S2 = 7 − 1 = 6 feet. That is the length of the bed.

Rule

Same y: distance = $∣ x_{2} - x_{1} ∣$ Same x: distance = $∣ y_{2} - y_{1} ∣$

The bars | |: absolute value

The two straight bars mean "the size of the number, ignoring any minus sign". So |7| = 7 and |−7| = 7. We use them because a distance can never be negative. If you subtract in the "wrong" order and get −7, the bars turn it back into 7.

1.4Distance between any two points

Now the harder case. What if the two points are not on the same row or column? Then the straight line joining them is slanted, and we cannot just count squares along it.

The trick is to use a result you learnt in Class 8.

Baudhāyana-Pythagoras theorem

In a right-angled triangle, the longest side (called the hypotenuse, opposite the right angle) is related to the two shorter sides like this:

(longest side)² = (one short side)² + (other short side)²

So: longest side = √(one short side² + other short side²).

Think of a shortcut across a field

Suppose you want to cross a rectangular field from one corner to the opposite corner. You could walk along two edges: first across, then up. Or you could cut straight across the middle. The straight path is the hypotenuse of a right triangle, and the two edges are the short sides. The theorem tells you exactly how long the shortcut is.

A worked example from the book: triangle ADM

The book looks at a triangle with corners A (3, 4), D (7, 1) and M (9, 6). Let us find the length of side AD.

Redrawn Figs. 1.6 and 1.7. To find AD, we add a helper point C (3, 1) straight below A and level with D. Now ACD is a right-angled triangle, with the right angle at C.

Finding AD step by step

Walk across. From A to D, the x-coordinate changes from 3 to 7. Distance across = 7 − 3 = 4. This is side CD.
Walk up or down. The y-coordinate changes from 4 to 1. Distance down = 4 − 1 = 3. This is side AC.
Use the theorem. AD is the hypotenuse, so AD² = 4² + 3² = 16 + 9 = 25.
Take the square root. AD = √25 = 5 units.

Using exactly the same steps for the other two sides:

DM: from D (7, 1) to M (9, 6), across = 9 − 7 = 2, up = 6 − 1 = 5. So DM = √(2² + 5²) = √(4 + 25) = √29 units, which is about 5.39.
MA: from M (9, 6) to A (3, 4), across = 9 − 3 = 6, down = 6 − 4 = 2. So MA = √(6² + 2²) = √(36 + 4) = √40 units, which is about 6.32.

Answers do not have to be whole numbers

√29 and √40 are not whole numbers, and that is completely fine. It is normal to leave the answer as √29, or to write an approximate decimal like 5.39.

The general rule: the distance formula

Let us write the same steps for any two points. Call them A (x₁, y₁) and D (x₂, y₂). The little numbers ₁ and ₂ just mean "first point" and "second point".

Redrawn Fig. 1.8. The helper point F is straight below A and level with D, so it has A's x-coordinate and D's y-coordinate.

Distance formula

$distance = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

In words: subtract the x's, subtract the y's, square both, add them, then take the square root.

Why the order of subtraction does not matter

If you subtract the "wrong" way round, you get a negative number, like −4 instead of 4. But then you square it, and (−4)² = 16, the same as 4² = 16. Squaring always removes the minus sign. So you can take the points in any order and still get the right distance. We are really just measuring how far you shift along each axis.

Your turn: find the distance between (−2, −3) and (4, 5).Tap to reveal

Across: 4 − (−2) = 4 + 2 = 6.
Up: 5 − (−3) = 5 + 3 = 8.
Distance² = 6² + 8² = 36 + 64 = 100.
Distance = √100 = 10 units.

1.4Reflection: points in a mirror

Imagine the y-axis is a mirror standing upright. If you reflect triangle ADM in this mirror, each corner jumps to the other side, the same distance away from the mirror.

Redrawn Fig. 1.9. The original triangle is on the right. Its reflection in the y-axis is on the left. The little mark ′ (read "dash" or "prime") is how we name the reflected point: A′ is the image of A.

Look at what happened to the coordinates:

Original	Reflected in the y-axis
A (3, 4)	A′ (−3, 4)
D (7, 1)	D′ (−7, 1)
M (9, 6)	M′ (−9, 6)

The reflection rules

Reflect in the y-axis: the x-coordinate changes sign, the y-coordinate stays the same. So (x, y) becomes (−x, y).

Reflect in the x-axis: the y-coordinate changes sign, the x-coordinate stays the same. So (x, y) becomes (x, −y).

Do the side lengths change?

Let us check side A′D′, using the helper point C′ (−3, 1):

Across: C′D′ = −3 − (−7) = −3 + 7 = 4
Up: A′C′ = 4 − 1 = 3
A′D′ = √(4² + 3²) = √25 = 5 units, exactly the same as AD.

In the same way, D′M′ = √((−2)² + 5²) = √29 and M′A′ = √((−6)² + 2²) = √40. The negative numbers get squared, so they become positive. Every side has the same length as before.

Big result

A reflection keeps all lengths the same. The shape and size of the triangle do not change. Only its position changes, and it gets flipped like a mirror image.

Think and reflect: what stayed the same, and what changed?Tap to reveal

Stayed the same: the lengths of all three sides, the angles, the shape and the size. Also, every y-coordinate.

Changed: the position (it moved to the other side of the y-axis), the sign of every x-coordinate, and the direction the triangle faces (it is flipped, the way your right hand looks like a left hand in a mirror).

Would the same be true if we reflected the triangle in the x-axis instead?Tap to reveal

Yes. The lengths, angles, shape and size would again stay the same. The difference is which coordinate changes sign: now the y-coordinates change sign and the x-coordinates stay. For example, A (3, 4) would become (3, −4), and the triangle would appear flipped upside down below the x-axis, in Quadrant IV.

ExEnd-of-chapter exercises, solved

Try each question on paper first. Then open it to check your answer and see the full reasoning. Questions marked with a star (*) are harder, and some of them teach you new ideas, like the midpoint of a line.

1. What are the coordinates of the point where the two axes cross?Tap to reveal

That point is the origin O. Its x-coordinate is 0 and its y-coordinate is 0, so it is (0, 0).

2. Point W has x-coordinate −5. H is on the line through W parallel to the y-axis. What can you say about H's coordinates? Which quadrants can H be in?Tap to reveal

A line parallel to the y-axis is a straight up-and-down line. Every point on it is the same distance left or right of O. W is at x = −5, so every point on this line has x = −5.

So H = (−5, y), where y can be any number.

If y is positive, H is left and up: Quadrant II.
If y is negative, H is left and down: Quadrant III.
If y = 0, H is (−5, 0), which is on the x-axis and in no quadrant.

3. R (3, 0), A (0, −2), M (−5, −2), P (−5, 2) are joined in order. Predict two perpendicular sides, a side parallel to an axis, and two points that are mirror images.Tap to reveal

Look for points that share a coordinate.

A (0, −2) and M (−5, −2) share y = −2, so side AM is horizontal.
M (−5, −2) and P (−5, 2) share x = −5, so side MP is vertical.

(i) Perpendicular sides: AM and MP. A horizontal line and a vertical line always meet at a right angle.

(ii) Side parallel to an axis: AM is parallel to the x-axis. (MP is also parallel to the y-axis.)

(iii) Mirror images: M (−5, −2) and P (−5, 2). They have the same x, and their y's are opposites. So they are mirror images of each other in the x-axis.

Plotting RAMP confirms the predictions: the right angle is at M.

4. Plot Z (5, −6). Build a right-angled triangle IZN and find its side lengths.Tap to reveal

There are many correct answers. Here is one easy choice: put I at the origin (0, 0) and N at (5, 0). Then N is straight above Z, so the right angle is at N.

IN: along the x-axis from 0 to 5, so IN = 5.
NZ: straight down from y = 0 to y = −6, so NZ = 6.
IZ is the hypotenuse: IZ = √(5² + 6²) = √(25 + 36) = √61, about 7.81.

5. What would coordinates be like without negative numbers? Could we still locate every point?Tap to reveal

Without negative numbers, we could only describe points to the right of and above the origin. That means only Quadrant I and the positive parts of the axes. Points to the left or below O would have no address.

You could move the origin to a far corner (computer screens do this, putting (0, 0) at a corner), but that only covers a limited area. The whole plane goes on forever in every direction, so to label every point we need negative numbers. This is why Brahmagupta's work on negative numbers was so important.

*6. Are M (−3, −4), A (0, 0) and G (6, 8) on the same straight line? Find a way to check without drawing.Tap to reveal

Method 1: distances. If three points are on one straight line, the two short pieces add up exactly to the long piece.

MA = √(3² + 4²) = √25 = 5
AG = √(6² + 8²) = √100 = 10
MG = √(9² + 12²) = √(81 + 144) = √225 = 15

MA + AG = 5 + 10 = 15 = MG. So yes, they are on one straight line, with A in the middle.

Method 2: steps. From M to A you go 3 across and 4 up. From A to G you go 6 across and 8 up, which is the same "3 across for every 4 up" pattern, just doubled. Keeping the same steepness means you are still on the same straight line.

*7. Use your method to check R (−5, −1), B (−2, −5) and C (4, −12).Tap to reveal

Steps method. From R to B: 3 across and 4 down. That is 4 down for every 3 across, about 1.33 down per step across. From B to C: 6 across and 7 down. That is about 1.17 down per step across. The steepness is different, so the line bends at B. They are not on one straight line.

Distance method. RB = √(9 + 16) = 5. BC = √(36 + 49) = √85 ≈ 9.220. RC = √(81 + 121) = √202 ≈ 14.213. RB + BC ≈ 14.220, which is slightly more than RC. So again, not on a straight line.

Why calculating beats drawing here

The difference is only about 0.007 units. On a drawing, these points would look like they are on a straight line. This is exactly why a calculation method is more reliable than just plotting.

*8. Using the origin as one corner, plot (i) a right-angled isosceles triangle, and (ii) an isosceles triangle with one corner in Quadrant III and one in Quadrant IV.Tap to reveal

"Isosceles" means two sides are equal. Many answers are possible. Here are simple ones.

(i) O (0, 0), A (4, 0), B (0, 4). OA runs along the x-axis and OB runs along the y-axis, so the angle at O is a right angle. OA = OB = 4, so it is isosceles.

(ii) O (0, 0), P (−3, −4) in Quadrant III, Q (3, −4) in Quadrant IV. OP = √(9 + 16) = 5 and OQ = √(9 + 16) = 5. Two equal sides, so it is isosceles. P and Q are mirror images in the y-axis, which is why the triangle is balanced.

*9. Is M the midpoint of ST in each case? What is the connection between the coordinates of M, S and T?Tap to reveal

The midpoint is the point exactly halfway between S and T. Halfway in x means the average of the two x-coordinates. Halfway in y means the average of the two y-coordinates.

S	M	T	True halfway point	Is M the midpoint?
(−3, 0)	(0, 0)	(3, 0)	((−3+3)/2, (0+0)/2) = (0, 0)	Yes
(2, 3)	(3, 4)	(4, 5)	((2+4)/2, (3+5)/2) = (3, 4)	Yes
(0, 0)	(0, 5)	(0, −10)	(0, (0−10)/2) = (0, −5)	No, SM = 5 but MT = 15
(−8, 7)	(0, −2)	(6, −3)	((−8+6)/2, (7−3)/2) = (−1, 2)	No

Midpoint formula

$M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$

In words: average the x's, and average the y's.

*10. M (−7, 1) is the midpoint of A (3, −4) and B (x, y). Find B.Tap to reveal

The x-coordinate of M is the average of A's x and B's x:

(3 + x) ÷ 2 = −7, so 3 + x = −14, so x = −17.
(−4 + y) ÷ 2 = 1, so −4 + y = 2, so y = 6.

B = (−17, 6)

Check: from A to M you move −10 across and +5 up. From M to B you move another −10 across and +5 up. Equal moves, so M really is in the middle.

*11. P and Q split AB into three equal parts (P nearer A). Find P and Q for A (4, 7) and B (16, −2).Tap to reveal

The whole journey from A to B is: across 16 − 4 = 12, and down 7 − (−2) = 9.

Split into three equal parts, each part is 12 ÷ 3 = 4 across and 9 ÷ 3 = 3 down.

P = A plus one part = (4 + 4, 7 − 3) = (8, 4)
Q = A plus two parts = (4 + 8, 7 − 6) = (12, 1)

Checking with midpoints, as the book suggests: P should be the midpoint of A and Q: ((4 + 12)/2, (7 + 1)/2) = (8, 4). Correct. Q should be the midpoint of P and B: ((8 + 16)/2, (4 − 2)/2) = (12, 1). Correct.

*12. Show that A (1, −8), B (−4, 7), C (−7, −4) lie on a circle K centred at O. Is D (−5, 6) or E (0, 9) inside, on, or outside K?Tap to reveal

A circle centred at O is the set of all points at the same distance from O. That distance is the radius. So we find each point's distance from O (0, 0).

OA = √(1² + 8²) = √(1 + 64) = √65
OB = √(4² + 7²) = √(16 + 49) = √65
OC = √(7² + 4²) = √(49 + 16) = √65

All three are √65 away from O, so they lie on one circle. Radius = √65, about 8.06.

D (−5, 6): OD = √(25 + 36) = √61. Since √61 is less than √65, D is inside the circle.

E (0, 9): OE = 9 = √81. Since √81 is more than √65, E is outside the circle.

*13. D (5, 1), E (6, 5) and F (0, 3) are the midpoints of the sides of triangle ABC. Find A, B and C.Tap to reveal

We take the usual naming: D is the midpoint of BC, E is the midpoint of CA, and F is the midpoint of AB.

A neat trick. "D is the midpoint of BC" means B + C = 2D (for x and y separately). Similarly, C + A = 2E and A + B = 2F. Adding all three: 2(A + B + C) = 2(D + E + F), so A + B + C = D + E + F.

D + E + F = (5 + 6 + 0, 1 + 5 + 3) = (11, 9). This is A + B + C.
A = (A + B + C) − (B + C) = (11, 9) − 2D = (11 − 10, 9 − 2) = (1, 7)
B = (11, 9) − 2E = (11 − 12, 9 − 10) = (−1, −1)
C = (11, 9) − 2F = (11 − 0, 9 − 6) = (11, 3)

Check: midpoint of B and C = ((−1 + 11)/2, (−1 + 3)/2) = (5, 1) = D. Midpoint of C and A = ((11 + 1)/2, (3 + 7)/2) = (6, 5) = E. Midpoint of A and B = ((1 − 1)/2, (7 − 1)/2) = (0, 3) = F. All correct.

14. City streets are 200 m apart, 10 in each direction. How many street crossings can be called (4, 3)? How many can be called (3, 4)?Tap to reveal

(i) The model. Using 1 cm = 200 m, each street is a line 1 cm from the next one. Draw 10 vertical lines (North-South streets) and 10 horizontal lines (East-West streets), 1 cm apart, with the two main roads crossing in the centre.

(ii) Naming crossings. The first number is the North-South street and the second number is the East-West street. If the streets in each direction are numbered 1 to 10 in one fixed order (for example, North-South streets counted from the west, East-West streets counted from the south), then:

(a) Exactly one crossing is (4, 3): where the 4th North-South street meets the 3rd East-West street.
(b) Exactly one crossing is (3, 4): where the 3rd North-South street meets the 4th East-West street.

These are two different crossings. This is the same lesson as before: the order of the pair matters.

A deeper thought

If instead you counted streets outward from the centre on both sides, "the 4th street" could mean 4th to the east or 4th to the west. Then (4, 3) could point to four different crossings. That confusion is exactly what negative numbers solve: east could be positive and west negative, north positive and south negative, just like our axes.

15. A screen is 800 by 600 pixels with (0, 0) at the bottom-left. Circle A: centre (100, 150), radius 80. Circle B: centre (250, 230), radius 100. Is any part off-screen? Do the circles cross?Tap to reveal

(i) Off-screen? Check how far each circle reaches in every direction:

Circle A: left edge 100 − 80 = 20, right edge 180, bottom 150 − 80 = 70, top 230. All between 0 and the screen limits.
Circle B: left edge 250 − 100 = 150, right edge 350, bottom 230 − 100 = 130, top 330. Also all inside.

So no part of either circle is off the screen.

(ii) Do they cross? Find the distance between the centres: across 250 − 100 = 150, up 230 − 150 = 80.

Distance = √(150² + 80²) = √(22500 + 6400) = √28900 = 170 pixels.

If the circles were just touching, the centres would be 80 + 100 = 180 apart. They are only 170 apart, which is closer than that, so the circles overlap and cross each other. (Neither circle is hidden inside the other, because 170 is much more than the difference 100 − 80 = 20.)

16. Plot A (2, 1), B (−1, 2), C (−2, −1), D (1, −2). Is ABCD a square? What is its area?Tap to reveal

Sides:

AB = √((2 − (−1))² + (1 − 2)²) = √(9 + 1) = √10
BC = √((−1 − (−2))² + (2 − (−1))²) = √(1 + 9) = √10
CD = √((−2 − 1)² + (−1 − (−2))²) = √(9 + 1) = √10
DA = √((1 − 2)² + (−2 − 1)²) = √(1 + 9) = √10

Diagonals: AC = √(4² + 2²) = √20 and BD = √(2² + 4²) = √20.

All four sides are equal and both diagonals are equal. A four-sided shape with equal sides could be a tilted rhombus, but a rhombus with equal diagonals must be a square. So yes, ABCD is a square. It is simply turned at an angle.

Area = side × side = √10 × √10 = 10 square units.

A square does not need to sit straight. This one is tilted, but all its sides and angles are equal.

RecapThe whole chapter on one screen

To locate a point on a flat surface, we use two number lines at right angles: the horizontal x-axis and the vertical y-axis.
They cross at the origin O (0, 0). The flat surface is called the Cartesian plane, the coordinate plane, or the xy-plane.
Right and up are positive. Left and down are negative.
A point's address is the ordered pair (x, y): first across, then up or down. The x-coordinate is the distance from the y-axis, and the y-coordinate is the distance from the x-axis.
Points on the x-axis look like (x, 0). Points on the y-axis look like (0, y).
The axes split the plane into four quadrants: I (+, +), II (−, +), III (−, −), IV (+, −). Points on an axis are in no quadrant.
(x, y) and (y, x) are the same point only when x = y.
Same y: distance = |x₂ − x₁|. Same x: distance = |y₂ − y₁|.
Any two points: distance = √((x₂ − x₁)² + (y₂ − y₁)²). This comes from the Baudhāyana-Pythagoras theorem.
Midpoint = (average of the x's, average of the y's).
Reflecting in the y-axis turns (x, y) into (−x, y). Reflecting in the x-axis turns (x, y) into (x, −y). Reflections keep all lengths the same.

WordsWords to know

Coordinates: The pair of numbers (x, y) that gives a point's exact position.
Ordered pair: A pair of numbers where the order matters. (2, 5) and (5, 2) are different.
x-coordinate: The first number. How far left (−) or right (+) the point is from the y-axis. Some books call it the abscissa.
y-coordinate: The second number. How far down (−) or up (+) the point is from the x-axis. Some books call it the ordinate.
Origin: The point (0, 0) where the axes cross.
Axis, axes: One of the two number lines; "axes" is the plural.
Quadrant: One of the four regions the axes divide the plane into.
Perpendicular: Meeting at a right angle (90°).
Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
Absolute value: The size of a number without its sign, written with bars: |−3| = 3.
Midpoint: The point exactly halfway between two points.
Reflection: A mirror image of a point or shape across a line.
Scale: How a length on a drawing compares to the real length, like 1 cm : 1 foot.

CheckQuick self-check

Answer in your head, then tap to see if you were right.

Which quadrant is (−4, 7) in?Answer

Quadrant II. x is negative (left) and y is positive (up).

A point is on the y-axis, 6 units below O. What are its coordinates?Answer

(0, −6). On the y-axis, x = 0. Below means negative.

Are (3, −2) and (−2, 3) the same point?Answer

No. (3, −2) is in Quadrant IV and (−2, 3) is in Quadrant II. The order matters.

What is the distance between (1, 2) and (1, 9)?Answer

7. Same x, so just subtract the y's: 9 − 2 = 7.

What is the distance from the origin to (6, 8)?Answer

10. √(6² + 8²) = √(36 + 64) = √100 = 10.

Reflect (5, −3) in the y-axis. Then reflect the original point in the x-axis.Answer

In the y-axis: (−5, −3). In the x-axis: (5, 3).

What is the midpoint of (2, 8) and (6, −4)?Answer

(4, 2). Average of x's: (2 + 6) ÷ 2 = 4. Average of y's: (8 − 4) ÷ 2 = 2.