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State Board · Class 9 · Mathematics Part 2

Lesson 7: Co-ordinate Geometry

4 concepts · 32 practice questions with answers · Free to read

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Axes, quadrants and co-ordinates of a point

From daily life

A cinema ticket says "row 4, seat 7". Two numbers fix the exact place. On a map too we can name a place using horizontal and vertical distances. In maths this method is called co-ordinates.

Simple definition

Two number lines meeting at right angles are the axes: the horizontal X-axis and the vertical Y-axis. They meet at the origin O (0, 0). In the point (x, y), x is the horizontal distance from the origin and y is the vertical distance. The two axes divide the plane into four quadrants.

The X-axis and Y-axis divide the plane into four quadrantsThe four quadrants and the signs of co-ordinatesXYO1st quadrant(+, +)2nd quadrant(−, +)3rd quadrant(−, −)4th quadrant(+, −)+X−X+Y−Y

In the figure: Right and up are positive; left and down are negative.

Live figure

Axes, quadrants and co-ordinates of a point

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Rule 1: Identify the quadrant from the signs — (+, +) first; (−, +) second; (−, −) third; (+, −) fourth. Quadrants are counted anticlockwise.

  • Easy: (4, 3): both positive ⇒ first quadrant.
  • Medium: (−5, 2): x negative, y positive ⇒ second. (−1, −6) ⇒ third. (7, −2) ⇒ fourth.
  • Exam type: (0, 5) is in no quadrant; it lies on the Y-axis. (−3, 0) lies on the X-axis. Points on the axes belong to no quadrant.

Rule 2: Reading a point: first x, then y — From the origin go x units sideways (right if positive), then y units vertically (up if positive). (3, 2) and (2, 3) are different points.

  • Easy: (3, 2): 3 to the right, 2 up.
  • Medium: (−2, 4): 2 left, 4 up. (−4, −3): 4 left, 3 down.
  • Exam type: (2, −3) and (−3, 2) fall in different quadrants (fourth and second). Changing the order changes the point.

Rule 3: Points on the axes — On the X-axis y = 0, so the point is (a, 0). On the Y-axis x = 0, so the point is (0, b). The origin (0, 0) lies on both axes.

  • Easy: (6, 0) is on the X-axis at distance 6.
  • Medium: (0, −4) is on the Y-axis, 4 units below the origin.
  • Exam type: Distance between (a, 0) and (0, a) for a = 3: points (3, 0) and (0, 3); right triangle 3, 3 ⇒ distance √18 = 3√2 ≈ 4.24.
Remember

x is horizontal and y is vertical. x first, then y. Quadrants run anticlockwise.

Common mistake

Treating (3, 2) and (2, 3) as the same. A different order is a different point.

Shortcut

Do not put an axis point in any quadrant: if one number is 0 the point lies on an axis. Memory clue: (x, y): "first across, then up"; x through the door, y up the stairs.

Spot the difference

QuadrantSign of xSign of y
First++
Second−+
Third−−
Fourth+−

Practice questions

Easy
  1. 1In which quadrant is (5, 7)?
  2. 2In which quadrant is (−3, −8)?
  3. 3Where is (0, 6)?
Medium
  1. 4In which quadrant is (6, −2)?
  2. 5What is the x-value of any point on the Y-axis?
  3. 6How do you reach (−4, 3) from the origin?
Challenging
  1. 7If the x and y of (−2, 5) are swapped, in which quadrant does the new point lie?
  2. 8The point (k, 0) is (−7, 0). Where is it on the X-axis and how far from the origin?
Show answers (try them yourself first)
  1. 1First (both positive). 2) Third (both negative). 3) On the Y-axis, 6 units above the origin.
  2. 4Fourth (x positive, y negative). 5) 0. 6) 4 to the left, 3 up.
  3. 7The new point is (5, −2): x positive, y negative ⇒ fourth quadrant. 8) 7 units to the left of the origin; distance 7.

Plotting points and distances

From daily life

On graph paper you count rows and columns and put a dot. Just like "e4" on a chess board, here too points are placed by counting squares.

Simple definition

To plot a point means to mark its place on graph paper from its co-ordinates. Each point has exactly one pair of co-ordinates and each pair gives exactly one point.

Points in the four quadrants and their co-ordinatesPlotting and reading points-5-4-3-2-112345-5-4-3-2-112345xyA(3, 2)B(-2, 4)C(-4, -3)D(2, -3)A (3, 2)3 to the right2 upx first, then y(3, 2) ≠ (2, 3)

In the figure: A(3, 2): 3 right and 2 up. Points of all four quadrants on one graph.

Rule 1: Steps to plot — (1) From the origin move x units horizontally. (2) From there move y units vertically. (3) Mark the dot and write its name.

  • Easy: P(2, 3): 2 right, 3 up.
  • Medium: Q(−3, −1): 3 left, 1 down. R(4, −2): 4 right, 2 down.
  • Exam type: The five points (1, 1), (2, 2), (3, 3), (−1, −1), (−2, −2) lie on a straight line because x = y in each.

Rule 2: Points with the same y: distance = difference of x — If both points have the same y the distance = bigger x − smaller x.

  • Easy: (2, 5) and (7, 5): distance 7 − 2 = 5.
  • Medium: (−4, 1) and (3, 1): distance 3 − (−4) = 7.
  • Exam type: (−6, −2) and (−1, −2): distance −1 − (−6) = 5. Even with negatives the distance comes out positive.

Rule 3: Points with the same x: distance = difference of y — If both points have the same x the distance = bigger y − smaller y.

  • Easy: (3, 1) and (3, 6): distance 6 − 1 = 5.
  • Medium: (−2, −3) and (−2, 4): distance 4 − (−3) = 7.
  • Exam type: The points (1, 2), (6, 2), (6, 9) form a right triangle: horizontal distance 5, vertical distance 7. Area ½ × 5 × 7 = 17.5.
Remember

On one horizontal line y is the same; on one vertical line x is the same. Ignore the equal number and take the difference of the other.

Common mistake

Errors in differences with negatives: the gap from −4 to 3 is not 3 − 4; it is 3 − (−4) = 7.

Shortcut

Sure way: count on the number line. 4 from −4 to 0 and 3 from 0 to 3 make 7. Memory clue: Distance: if x or y is the same, just find the difference.

Spot the difference

PositionWhat is equalDistance
On a horizontal lineydifference of x
On a vertical linexdifference of y
SlantingneitherPythagoras

Practice questions

Easy
  1. 1To plot (2, 4), how far right and how far up do you go?
  2. 2What is the distance between (1, 3) and (6, 3)?
  3. 3What is the distance between (5, 2) and (5, 9)?
Medium
  1. 4What is the distance between (−3, 2) and (4, 2)?
  2. 5What is the distance between (−1, −5) and (−1, 3)?
  3. 6What triangle do (0, 0), (4, 0), (4, 3) form? Find its area.
Challenging
  1. 7Find the hypotenuse of the triangle (0, 0), (5, 0), (5, 12).
  2. 8Find the area of the triangle A(−2, 1), B(3, 1), C(3, −4).
Show answers (try them yourself first)
  1. 12 right and 4 up. 2) 6 − 1 = 5. 3) 9 − 2 = 7.
  2. 44 − (−3) = 7. 5) 3 − (−5) = 8. 6) A right triangle (horizontal 4, vertical 3). Area ½ × 4 × 3 = 6.
  3. 7Horizontal 5, vertical 12 ⇒ hypotenuse √(25 + 144) = 13. 8) AB = 5, BC = 5, right angle at B. Area ½ × 5 × 5 = 12.5.

Lines parallel to the axes

From daily life

On a ground, a horizontal line and a vertical line can be drawn. Every point on the horizontal line is at the same height, and every point on the vertical line is at the same side distance.

Simple definition

The line y = b is parallel to the X-axis (horizontal); every point on it has y = b and any x. The line x = a is parallel to the Y-axis (vertical); every point on it has x = a and any y.

The lines y = 3, y = −4, x = −2 and x = 4Lines parallel to the axes-5-4-3-2-112345-5-4-3-2-112345xyy = 3y = −4x = −2x = 4y = b: horizontalparallel to the X-axisx = a: verticalparallel to the Y-axis

In the figure: y = 3 and y = −4 are horizontal; x = −2 and x = 4 are vertical.

Live figure

Lines parallel to the axes

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Rule 1: y = b is a horizontal line — Parallel to the X-axis at distance b. Above it if b is positive, below if negative. y = 0 is the X-axis itself.

  • Easy: y = 3: the points (−2, 3), (0, 3), (5, 3) all lie on this line.
  • Medium: y = −4: a horizontal line 4 units below the X-axis.
  • Exam type: The given point is (6, −2) and the line is parallel to the X-axis ⇒ y = −2.

Rule 2: x = a is a vertical line — Parallel to the Y-axis at distance a. To the right if a is positive, to the left if negative. x = 0 is the Y-axis itself.

  • Easy: x = 4: the points (4, −3), (4, 0), (4, 6) lie on this line.
  • Medium: x = −2: a vertical line 2 units to the left of the Y-axis.
  • Exam type: The line through (−5, 8) parallel to the Y-axis ⇒ x = −5.

Rule 3: Meeting point of two lines — A horizontal line y = b and a vertical line x = a meet at (a, b). Two horizontal lines, or two vertical lines, never meet.

  • Easy: x = 4 and y = 3 ⇒ meeting point (4, 3).
  • Medium: x = −2 and y = −4 ⇒ (−2, −4).
  • Exam type: x = 4 and x = −2 are parallel; the gap between them is 4 − (−2) = 6.
Remember

y = b ⇒ horizontal (parallel to the X-axis). x = a ⇒ vertical (parallel to the Y-axis).

Common mistake

Thinking x = 3 is horizontal. x = 3 means x is always 3, so it is vertical.

Shortcut

Remember: in y = b the y does not change, so the height is the same and the line is horizontal. Memory clue: x = a is a vertical line, y = b is a horizontal line: "y means flat".

Spot the difference

y = bx = a
Directionhorizontalvertical
Parallel toX-axisY-axis
Points(anything, b)(a, anything)

Practice questions

Easy
  1. 1Is the line y = 5 horizontal or vertical?
  2. 2The line x = −3 is parallel to which axis?
  3. 3Does the point (2, 9) lie on the line x = 2?
Medium
  1. 4Find the meeting point of x = 6 and y = −2.
  2. 5Write the equation of the line through (−4, 7) parallel to the X-axis.
  3. 6What is the distance between x = 5 and x = −1?
Challenging
  1. 7Find the distance between y = 4 and y = −3. How is it cut by any vertical line?
  2. 8If (k, 5) lies on x = 2, what is k? If it lies on y = 5, what can k be?
Show answers (try them yourself first)
  1. 1Horizontal. 2) The Y-axis. 3) Yes, because x = 2.
  2. 4(6, −2). 5) y = 7. 6) 5 − (−1) = 6.
  3. 74 − (−3) = 7. A vertical line cuts each line at one point and the distance between those two points is 7. 8) On x = 2, k = 2. On y = 5, k can be anything.

Rectangles and areas from lines

From daily life

If ropes are tied on all four sides of a field, two horizontal and two vertical ropes make a rectangular plot. Its length and width come from the gaps between the ropes.

Simple definition

Two horizontal lines y = b₁, y = b₂ and two vertical lines x = a₁, x = a₂ together make a rectangle. Length = difference of x, width = difference of y. Area = length × width and perimeter = 2 × (length + width).

Rectangle formed by x = 1, x = 5, y = 1, y = 4A rectangle made by four lines1234567123456xyP(1,1)Q(5,1)R(5,4)S(1,4)x = 1, x = 5y = 1, y = 4Length 5 − 1 = 4Width 4 − 1 = 3Area = 4 × 3 = 12

In the figure: The lines x = 1, x = 5, y = 1, y = 4 give vertices P(1,1), Q(5,1), R(5,4), S(1,4). Length 4, width 3.

Rule 1: Find the vertices — Each vertical line meets each horizontal line; the four meeting points are the four vertices.

  • Easy: x = 1, x = 5 and y = 1, y = 4 ⇒ (1,1), (5,1), (5,4), (1,4).
  • Medium: x = −2, x = 3 and y = 0, y = 5 ⇒ (−2,0), (3,0), (3,5), (−2,5).
  • Exam type: x = −3, x = −1 and y = −2, y = 4 ⇒ (−3,−2), (−1,−2), (−1,4), (−3,4).

Rule 2: Length, width, area — Length = |a₂ − a₁|, width = |b₂ − b₁|. Area = length × width.

  • Easy: In the figure length 5 − 1 = 4, width 4 − 1 = 3, area 12.
  • Medium: x = −2, 3 and y = 0, 5: length 5, width 5, area 25 (this is a square).
  • Exam type: x = −3, −1 and y = −2, 4: length 2, width 6, area 12, perimeter 2 × (2 + 6) = 16.

Rule 3: Diagonal and centre — The diagonal of the rectangle comes from Pythagoras: √(length² + width²). The centre = the average of the x values and of the y values.

  • Easy: Length 4, width 3 ⇒ diagonal √(16 + 9) = 5.
  • Medium: In the figure the centre is (1 + 5) ÷ 2 = 3 and (1 + 4) ÷ 2 = 2.5 ⇒ (3, 2.5).
  • Exam type: Length 5, width 12 ⇒ diagonal 13. Centre if P(0,0) and R(5,12): (2.5, 6).
Remember

The x-difference of the two vertical lines is the length; the y-difference of the two horizontal lines is the width. The area is their product.

Common mistake

Getting a negative length or width. Always take bigger − smaller so the distance is positive.

Shortcut

If length and width are equal, the shape is a square. Memory clue: Area of a rectangle = length × breadth; both come from differences of co-ordinates.

Spot the difference

NeededFormulaIn the figure
Lengthdifference of x4
Widthdifference of y3
Arealength × width12
Perimeter2 × (length + width)14

Practice questions

Easy
  1. 1Find the length of the rectangle made by x = 2, x = 6, y = 1, y = 4.
  2. 2Find the width of the same rectangle.
  3. 3Find the area of the same rectangle.
Medium
  1. 4x = −1, x = 4 and y = 2, y = 5: find the area.
  2. 5What shape do x = 0, x = 5 and y = 0, y = 5 make?
  3. 6Find the perimeter of the rectangle from x = 1, x = 5, y = 1, y = 4.
Challenging
  1. 7Find the area and the diagonal of the rectangle from x = −2, x = 3 and y = −1, y = 3.
  2. 8A rectangle has length 6 and width 8. Find the diagonal and the centre if one vertex is (0, 0) and the opposite one is (6, 8).
Show answers (try them yourself first)
  1. 16 − 2 = 4. 2) 4 − 1 = 3. 3) 4 × 3 = 12.
  2. 4Length 5, width 3 ⇒ 15. 5) A square (length = width = 5). 6) 2 × (4 + 3) = 14.
  3. 7Length 5, width 4, area 20, diagonal √(25 + 16) = √41 ≈ 6.4. 8) Diagonal √(36 + 64) = 10, centre (3, 4).