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CBSE · Class 9 · Mathematics

Chapter 1: Coordinates

5 concepts · 40 practice questions with answers · Free to read

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Ordered pairs and locating a point

From daily life

A seat in a cinema is found with a row and a seat number, like "Row 4, Seat 7". Two numbers in the right order fix one place. A map grid works the same way.

Simple definition

Two number lines cross at right angles at the origin O (0, 0): the horizontal x-axis and the vertical y-axis. A point is written as an ordered pair (x, y). The first number is the x-coordinate (how far right or left), the second is the y-coordinate (how far up or down).

Formula: Point P(x, y): move x along the x-axis first, then y parallel to the y-axis.

xy−4−3−2−11234−11234P(3,2)Q(-2,3)O

In the figure: P is 3 units right and 2 units up, so P(3, 2). Q is 2 units left and 3 up, so Q(−2, 3).

Rule 1: Order matters — (x, y) and (y, x) are different points unless x = y. Always write the x-coordinate first.

  • Easy: (2, 5) is 2 right, 5 up; (5, 2) is 5 right, 2 up.
  • Medium: (4, 4) is the same when reversed.
  • Exam-type: Origin: (0, 0). On the x-axis: (a, 0). On the y-axis: (0, b).

Rule 2: Reading a point from the graph — Drop a line to the x-axis for x, and a line to the y-axis for y.

  • Easy: A point 3 left of the y-axis and 1 above the x-axis is (−3, 1).
  • Medium: A point 4 below the x-axis on the y-axis is (0, −4).
  • Exam-type: A shop is 6 blocks east and 2 blocks south of the school at the origin: (6, −2).
Remember

x first (across), y second (up or down). Left and down are negative.

Common mistake

Plotting (3, 2) as 3 up and 2 right.

Memory clue

"Run before you jump": x (run) then y (jump).

Quick table

PointWhere
(3, 0)on x-axis
(0, −2)on y-axis
(0, 0)origin

Practice questions

Easy
  1. 1What are the coordinates of the origin?
  2. 2What is the y-coordinate (ordinate) of the point (4, 5)?
  3. 3A bee is at the point (−4, 2). It flies 5 units to the left and 4 units up. What is its new position?
Medium
  1. 4A bee is at the point (1, −6). It flies 3 units to the left and 4 units up. What is its new position?
  2. 5A point in the first quadrant is 2 units away from the y-axis and 7 units away from the x-axis. What are its coordinates?
  3. 6On a grid map, Asha's house is at (6, 5). Lata's house is 4 units to the right of it and 1 units below it. What are the coordinates of Lata's house?
Challenging
  1. 7Three vertices of a rectangle with sides parallel to the axes are (0, 0), (6, 0) and (6, 9). Find the fourth vertex.
  2. 8The ordered pairs (x + 1, 5) and (4, y + 2) represent the same point. Find x + y.
Show answers (try them yourself first)
  1. 1(0, 0)
  2. 25
  3. 3(−9, 6)
  4. 4(−2, −2)
  5. 5(2, 7)
  6. 6(10, 4)
  7. 7(0, 9)
  8. 86
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Ordered pairs and locating a point

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Quadrants and signs of coordinates

From daily life

A compass splits a city map into four parts: north-east, north-west, south-west and south-east. The axes split a graph in the same way.

Simple definition

The two axes divide the plane into four quadrants, numbered I, II, III, IV anticlockwise from the top right. Points on an axis lie in no quadrant.

Formula: I: (+, +) II: (−, +) III: (−, −) IV: (+, −)

xy−4−3−2−11234−4−3−2−11234IIIIIIIV(+, +)(−, +)(−, −)(+, −)

In the figure: Quadrant I is top right; numbering goes anticlockwise to IV at bottom right.

Rule 1: Sign tells the quadrant — Look at the sign of x first, then y.

  • Easy: (5, 3): both positive, so quadrant I.
  • Medium: (−2, 7): x negative, y positive, so quadrant II.
  • Exam-type: (−1, −8) is in III and (4, −6) is in IV.

Rule 2: Points on the axes — If x = 0 or y = 0 the point is on an axis, not in a quadrant.

  • Easy: (0, 5) is on the y-axis.
  • Medium: (−3, 0) is on the x-axis.
  • Exam-type: If xy > 0 the point is in quadrant I or III; if xy < 0, in II or IV.
Remember

Quadrants go anticlockwise. The product xy is positive in I and III, negative in II and IV.

Common mistake

Saying (0, 4) lies in quadrant I.

Memory clue

Go anticlockwise: (+,+), (−,+), (−,−), (+,−).

Quick table

Quadrantxy
I++
II−+
III−−
IV+−

Practice questions

Easy
  1. 1Into how many regions (quadrants) do the two axes divide the plane?
  2. 2In which quadrant does the point (−3, 1) lie?
  3. 3Identify the point that belongs to the first quadrant.
Medium
  1. 4In which quadrant does the point (−4.5, 4.5) lie?
  2. 5How many of the points (2, 4), (−7, 8), (4, 8), (8, 2) lie in the first quadrant?
  3. 6On which axis does the point (0, 8) lie?
Challenging
  1. 7The point (p, q) lies in the fourth quadrant. In which quadrant does the point (q, −p) lie?
  2. 8A point (x, y) has x·y > 0 and x + y < 0. In which quadrant does it lie?
Show answers (try them yourself first)
  1. 14
  2. 2second quadrant
  3. 3(2, 8)
  4. 4second quadrant
  5. 53
  6. 6y-axis
  7. 7third quadrant
  8. 8third quadrant
Live figure

Quadrants and signs of coordinates

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Axes, reflections and distance from the axes

From daily life

Your face in a mirror is the same distance behind the glass as you are in front. A point reflected in an axis behaves like that.

Simple definition

The distance of P(x, y) from the y-axis is |x| and from the x-axis is |y|. Reflecting in the x-axis changes the sign of y; reflecting in the y-axis changes the sign of x.

Formula: Reflect in x-axis: (x, y) → (x, −y). In y-axis: (x, y) → (−x, y). In origin: (x, y) → (−x, −y).

xy−4−3−2−11234−3−2−1123P(3,2)P'(3,-2)P''(-3,2)

In the figure: P′ is the mirror image of P in the x-axis; P″ is the mirror image in the y-axis.

Rule 1: Distance from an axis — Distance is never negative, so use the absolute value of the other coordinate.

  • Easy: (−5, 3) is 5 from the y-axis and 3 from the x-axis.
  • Medium: A point 4 from the x-axis in quadrant III has y = −4.
  • Exam-type: A point on the x-axis has y = 0, so its distance from the x-axis is 0.

Rule 2: Reflections — Keep the coordinate along the mirror line; flip the sign of the other one.

  • Easy: (4, −1) in the x-axis → (4, 1).
  • Medium: (−2, 6) in the y-axis → (2, 6).
  • Exam-type: A point and its reflection in the x-axis are 2|y| apart.
Remember

Distance from the x-axis is |y|; from the y-axis is |x|. Reflecting twice (in both axes) gives the reflection in the origin.

Common mistake

Mixing the axes: the distance from the x-axis uses y, not x.

Memory clue

"Reflect in x-axis: y flips. In y-axis: x flips."

Quick table

Mirror(3, 2) becomes
x-axis(3, −2)
y-axis(−3, 2)
origin(−3, −2)

Practice questions

Easy
  1. 1What is the distance of the point (6, 7) from the x-axis?
  2. 2What is the distance of the point (−5, −3) from the y-axis?
  3. 3What is the mirror image (reflection) of the point (−1, 3) in the x-axis?
Medium
  1. 4What is the mirror image (reflection) of the point (1, 7) in the y-axis?
  2. 5The point (−3, −7) is reflected through the origin (turned by a half-turn about O). Where does it land?
  3. 6Write the coordinates of the point that lies on the x-axis, 7 units to the right of the origin.
Challenging
  1. 7The point (−9, −6) is first reflected in the x-axis and the image is then reflected in the y-axis. What is the final position?
  2. 8The vertical line x = 6 acts like a mirror. What is the mirror image of the point (3, 0) in this line?
Show answers (try them yourself first)
  1. 17
  2. 25
  3. 3(−1, −3)
  4. 4(−1, 7)
  5. 5(3, 7)
  6. 6(7, 0)
  7. 7(9, 6)
  8. 8(9, 0)
Live figure

Axes, reflections and distance from the axes

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Distance between two points

From daily life

A kite flies 3 m east and 4 m north of you. The string is the shortest straight line to it: 5 m. A drone uses the same idea to find its straight-line distance.

Simple definition

Join two points and draw a right triangle with horizontal and vertical sides. The distance is the hypotenuse, found with the Pythagoras theorem.

Formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]. Same y: d = |x₂ − x₁|. Same x: d = |y₂ − y₁|. From origin: d = √(x² + y²).

xy−1123456−112345A(1,1)B(5,4)C435

In the figure: Legs 4 and 3 give hypotenuse 5, so AB = 5.

Rule 1: Horizontal and vertical lines — When the points share a coordinate, just subtract the other one.

  • Easy: (2, 3) and (9, 3): d = |9 − 2| = 7.
  • Medium: (−4, 1) and (−4, −5): d = |1 − (−5)| = 6.
  • Exam-type: Both must have the same x or same y; otherwise use the formula.

Rule 2: Slanting lines — Find the horizontal and vertical gaps, square both, add, take the square root.

  • Easy: (1, 2) to (4, 6): √(3² + 4²) = 5.
  • Medium: (−2, −1) to (4, 7): √(36 + 64) = 10.
  • Exam-type: Origin to (5, 12): √(25 + 144) = 13.
Remember

Squares are never negative, so the order of the points does not matter. Pythagorean triples (3,4,5), (5,12,13), (8,15,17) save time.

Common mistake

Forgetting the square root, or writing (x₂ − x₁)² + (y₂ − y₁)² as the distance.

Memory clue

"Gap across, gap up, square, add, root."

Quick table

PointsDistance
(0, 0), (6, 8)10
(1, 1), (1, 9)8
(2, 3), (5, 7)5

Practice questions

Easy
  1. 1Which expression gives the distance between (x₁, y₁) and (x₂, y₂)?
  2. 2What is the distance between the points (−3, −2) and (−3, 1)?
  3. 3Find the distance between A(−5, 3) and B(2, 27).
Medium
  1. 4What is the distance of the point (−5, 6) from the origin?
  2. 5Find the distance between P(4, 4) and Q(2, −2).
  3. 6The distance between (k, −3) and (−2, −3) is 9. Which values can k take?
Challenging
  1. 7A is the point (3, −1) and B is the point (11, k), with B above A. If AB = 17, find k.
  2. 8Gauri walks 5 km east and then 12 km north from the school gate. How far is Gauri now from the gate in a straight line (in km)?
Show answers (try them yourself first)
  1. 1√[(x₂ − x₁)² + (y₂ − y₁)²]
  2. 23
  3. 325
  4. 4√61
  5. 5√40
  6. 6−11 or 7
  7. 714
  8. 813

Using the distance formula

From daily life

A surveyor checks whether three poles stand in one straight line by measuring the gaps between them. If two gaps add to the third, they do.

Simple definition

The distance formula helps to test collinear points, to name the type of a triangle or quadrilateral from its side lengths, and to find a point that is equidistant from two given points.

Formula: Collinear: AB + BC = AC. Right angle at B: AB² + BC² = AC². Equidistant from A and B: PA = PB.

xy−4−3−2−1123−2−1123A(-3,-1)B(1,-1)C(1,2)

In the figure: AB = 4, BC = 3 and AC = 5, and 4² + 3² = 5²: a right angle at B.

Rule 1: Collinear points — Find all three distances. If the two smaller ones add to the biggest, the points lie on one line.

  • Easy: (0,0), (3,4), (6,8): 5 + 5 = 10, so collinear.
  • Medium: (0,0), (1,1), (2,3): √2 + √5 ≠ √13, not collinear.
  • Exam-type: Never add square roots by guessing; compare squared values carefully.

Rule 2: Type of triangle — Equal sides: isosceles/equilateral. Pythagoras fits: right-angled.

  • Easy: A(1,1), B(5,1), C(3,4): AC = BC = √13, isosceles.
  • Medium: A(0,0), B(4,0), C(4,3): 16 + 9 = 25, right angle at B.
  • Exam-type: Square: four equal sides and equal diagonals.

Rule 3: Equidistant points — Set PA² = PB², square-terms cancel and a simple equation remains.

  • Easy: To find (x, 0) equidistant from (2,3) and (6,5), set (x−2)²+9 = (x−6)²+25.
  • Medium: Expanding: −4x + 13 = −12x + 61, so 8x = 48 and x = 6.
  • Exam-type: The point (6, 0) is on the perpendicular bisector of the two points.
Remember

Compare squares first; take roots only at the end. A triangle with two equal sides is isosceles.

Common mistake

Declaring a right angle without checking that the squares add up exactly.

Memory clue

"Short + short = long: points in a line."

Quick table

TestCondition
CollinearAB + BC = AC
Right angleAB² + BC² = AC²
EquidistantPA = PB

Practice questions

Easy
  1. 1Find the perimeter of the triangle with vertices (0, 0), (6, 0) and (0, 8).
  2. 2Find the area (in square units) of the triangle with vertices (0, 0), (10, 0) and (10, 3).
  3. 3A rectangle has vertices (0, 0), (3, 0), (3, 4) and (0, 4). Find the length of its diagonal.
Medium
  1. 4What kind of triangle has the vertices A(−1, −1), B(3, −1) and C(−1, 3)?
  2. 5Find the perimeter of the triangle with vertices (4, −1), (7, −1) and (7, 3).
  3. 6Find the point on the x-axis that is equidistant from A(0, −4) and B(6, 2).
Challenging
  1. 7Find the point on the x-axis that is equidistant from A(−2, −4) and B(4, 2).
  2. 8The point (0, y) is 13 units from (−5, 0). Find the positive value of y.
Show answers (try them yourself first)
  1. 124
  2. 215
  3. 35
  4. 4right-angled and isosceles
  5. 512
  6. 6(2, 0)
  7. 7(0, 0)
  8. 812