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Orienting Yourself: The Use of Coordinates — Class 9 Mathematics

Ganita Manjari · Chapter 1

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Chapter summary

This chapter introduces the two-dimensional Cartesian coordinate system, tracing its rich history from the grid-planned streets of the Sindhu-Sarasvati civilisation through Brahmagupta's formalisation of zero and negative numbers to Descartes. Two perpendicular number lines — the horizontal -axis and the vertical -axis — meet at the origin , dividing the plane into four quadrants. Every point is named by an ordered pair , where is its perpendicular distance from the -axis and its distance from the -axis. Signs of the coordinates fix the quadrant: , , and for quadrants I to IV. Points on the -axis have the form and points on the -axis the form . For points along a line parallel to an axis, distance is just the absolute difference of one coordinate. For any two points the chapter derives the distance formula using the Baudhayana-Pythagoras theorem: . The midpoint of a segment is found by averaging coordinates.

Cartesian plane and axesQuadrants and signsCoordinates of a pointDistance formulaMidpoint of a segment

Key ideas & terms

Origin: The point where the -axis and -axis intersect, having coordinates .
Quadrant: One of the four regions into which the coordinate axes divide the Cartesian plane.
Abscissa: The -coordinate of a point, its perpendicular distance from the -axis.
Ordinate: The -coordinate of a point, its perpendicular distance from the -axis.
Distance formula: The length between and , equal to .
Midpoint: The point halfway along a segment, with coordinates .

Extra questions & answers

What are the coordinates of the origin?

The origin is ; both the - and -coordinate are zero.

In which quadrant does the point lie?

It lies in Quadrant II, where the -coordinate is negative and the -coordinate is positive.

What is the -coordinate of every point on the -axis?

It is always ; points on the -axis have the form .

Find the distance between and .

Using the distance formula, units.

When is the point the same as ?

Only when ; if then .

How do you find the distance between two points lying on a horizontal line?

Take the absolute value of the difference of their -coordinates, .

What are the coordinates of the midpoint of and ?

Averaging gives , the origin.

Why can three collinear points not all be vertices of a right triangle on the plane?

Collinear points lie on one straight line, so they cannot enclose a triangle of any kind.

Point is the midpoint of and ; if , find .

Since , solving gives .

How can you check whether three points are collinear without plotting?

Verify that the distance between the two outer points equals the sum of the two shorter distances, or that the slopes between pairs are equal.

Exercise Set 1.1

(i) If D1 R1 represents the door to Reiaan’s room, how far is the door from the left wall (the y-axis) of the room? How far is the door from the x-axis?

The door lies on the bottom edge of the room, which is the -axis. So its distance from the -axis is ft. From Fig. 1.3 the left end of the door is at . Hence the door is about ft from the left wall, i.e. the -axis. Every point of the door has and between about and .

(ii) What are the coordinates of D1 ?

is on the -axis so its -coordinate is . Reading Fig. 1.3, . If the door is taken as ft wide with , then exactly, because . In general where is its distance from the -axis.

(iii) If R1 is the point (11.5, 0), how wide is the door? Do you think this is a comfortable width for the room door? If a person in a wheelchair wants to enter the room, will he/she be able to do so easily?

Width = distance . With and , width ft cm. A standard interior door is about to ft cm \). So ft is a little narrow but usable for most people. For a wheelchair a clear opening of at least inches ft cm \) is recommended, so ft is tight and would not allow easy passage; a wider door of ft would be better.

(iv) If B1 (0, 1.5) and B2 (0, 4) represent the ends of the bathroom door, is the bathroom door narrower or wider than the room door?

and both lie on the -axis. Width = ft. This equals the room door width ft calculated above, so the two doors are equal in width. If is read slightly differently the widths are still very close, so the bathroom door is not wider than the room door; it is the same or marginally narrower/wider depending on the exact reading, about ft.

Think and Reflect

1. What are the standard widths for a room door? Look around your home and in school.

In India a common interior room door is about m to m wide, i.e. about ft to ft, and about m high. Main doors may be m wide. Measure with a tape: most classroom doors are close to m, bedroom doors about m. The ft door in the figure is therefore on the smaller side.

2. Are the doors in your school suitable for people in wheelchairs?

A wheelchair needs a clear width of at least mm inches . Many older school doors are only mm and have a step at the threshold, so they are not fully accessible. An accessible door should be at least mm wide, have no high sill, have lever handles and enough space to turn in front of it.

Think and Reflect

1. What is the x-coordinate of a point on the y-axis?

Every point on the -axis has no horizontal shift from the origin, so its perpendicular distance from the -axis is zero. Hence its -coordinate is . Any point on the -axis is of the form .

2. Is there a similar generalisation for a point on the x-axis?

Yes. Every point on the -axis has no vertical shift, so its -coordinate is . Any point on the -axis is of the form . The origin lies on both axes.

3. Does point Q (y, x) ever coincide with point P (x, y)? Justify your answer.

and coincide only when the two ordered pairs are equal, i.e. when . Then . If , the order matters and the points are reflections of each other in the line and do not coincide.

4. If x ≠ y, then (x, y) ≠ (y, x); and (x, y) = (y, x) if and only if x = y. Is this claim true?

Yes, it is true. Coordinates are ordered pairs. means and , which is equivalent to . So if the pairs are different, and if they are identical. For example but .

Exercise Set 1.2

1. Place Reiaan’s rectangular study table with three of its feet at the points (8, 9), (11, 9) and (11, 7). (i) Where will the fourth foot of the table be?

Let the three feet be , , . In a rectangle opposite sides are parallel to axes, so the fourth vertex completes the rectangle. The -coordinate must be to align with and the -coordinate must be to align with . Hence the fourth foot is .

1. Place Reiaan’s rectangular study table with three of its feet at the points (8, 9), (11, 9) and (11, 7). (ii) Is this a good spot for the table?

The points to lie in the top-right corner of the bedroom near the right wall and below the top wall, away from the bed and the doors. It gets light and leaves passage space, so it is a reasonable spot. Whether it is ideal depends on window and door swing, but it does not block the wardrobe or bathroom door.

1. Place Reiaan’s rectangular study table with three of its feet at the points (8, 9), (11, 9) and (11, 7). (iii) What is the width of the table? The length? Can you make out the height of the table?

Width = horizontal distance ft, length = vertical distance ft, so the table top is ft ft. The height cannot be read from a floor plan which is a 2-D projection on the -plane; height is the -coordinate perpendicular to the floor and is not shown.

2. If the bathroom door has a hinge at B1 and opens into the bedroom, will it hit the wardrobe? Are there any changes you would suggest if the door is made wider?

The bathroom door is on the -axis at from to . The wardrobe stands near the bottom wall with its top edge at about to and to . The door swings with radius ft into the room. The arc stays near the left wall and does not reach , so it will not hit the wardrobe in the present position. If the door were made wider, say ft or more, the swing would extend further and could foul the wardrobe or the bed. Suggestions: keep hinge at , make door open outward into the bathroom, or slide, or move wardrobe further along the bottom wall away from the bathroom wall.

3. Look at Reiaan’s bathroom. (i) What are the coordinates of the four corners O, F, R, and P of the bathroom?

Reading Fig. 1.5 with as origin, the bathroom is the rectangle to the left of the -axis. Its corners are approximately , , , . Dimensions are then ft along and ft along , which matches the labelled ft ft approximately, depending on scale reading; the key is that two corners are on the -axis with and two are units left with .

3. Look at Reiaan’s bathroom. (ii) What is the shape of the showering area SHWR in Reiaan’s bathroom? Write the coordinates of the four corners.

is a rectangle at the top of the bathroom. Taking the bathroom top as , , , , gives a ft ft shower. The sides and are parallel to the -axis and and parallel to the -axis.

3. Look at Reiaan’s bathroom. (iii) Mark off a 3 ft × 2 ft space for the washbasin and a 2 ft × 3 ft space for the toilet. Write the coordinates of the corners of these spaces.

One convenient allocation is: washbasin ft ft against the top-right of the bathroom, e.g. corners not overlapping the shower; and toilet ft ft at the bottom-left, e.g. corners . Any placement that stays inside , does not overlap , and has the stated side lengths measured parallel to axes is acceptable. The floor plan shows the washbasin near the top and the WC near the bottom left.

4. Other rooms in the house: (i) Reiaan’s room door leads from the dining room which has the length 18 ft and width 15 ft. The length of the dining room extends from point P to point A. Sketch the dining room and mark the coordinates of its corners.

Reiaan’s room occupies from to and from to . Point is the outer corner of the bathroom. The dining room extends further in negative or negative direction. Taking its length along the -direction from , a ft length and ft width could be for example corners , , , , so that ft along the -direction and the other side ft along negative . Any translation that keeps the stated dimensions and shares the side is correct; the key is to use negative coordinates beyond the bedroom.

4. Other rooms in the house: (ii) Place a rectangular 5 ft × 3 ft dining table precisely in the centre of the dining room. Write down the coordinates of the feet of the table.

The centre of a rectangle is the intersection of its diagonals, i.e. the midpoint of opposite corners. For the dining room above, centre . A ft ft table centred at has half-lengths ft and ft. Its four feet are at , i.e. for orientation with ft along . Swapping orientation gives . Any equivalent centred placement is acceptable.

Think and Reflect

1. In moving from A (3, 4) to D (7, 1), what distance has been covered along the x-axis? What about the distance along the y-axis?

Distance along -axis = absolute difference of -coordinates: units. Distance along -axis = absolute difference of -coordinates: units. These are the legs and of the right triangle used in Fig. 1.7.

2. Can these distances help you find the distance AD?

Yes. and are perpendicular, so triangle is right-angled at . By Baudhāyana-Pythagoras, units.

Think and Reflect

1. What has remained the same and what has changed with this reflection?

Reflection in the -axis preserves distances and shape. Side lengths remain units, and angles stay the same. What changes is the sign of the -coordinates: goes to . So , , . The figure moves from Quadrant I to Quadrant II.

2. Would these observations be the same if ΔADM is reflected in the x-axis (instead of the y-axis)?

Yes, in principle. Reflection in the -axis also is an isometry, so distances and angles are preserved. The change would be : the -coordinates change sign while stays same, moving the triangle to Quadrant IV. Lengths would still be .

End-of-Chapter Exercises

1. What are the x-coordinate and y-coordinate of the point of intersection of the two axes?

The axes intersect at the origin . Its -coordinate is and its -coordinate is .

2. Point W has x-coordinate equal to –5. Can you predict the coordinates of point H which is on the line through W parallel to the y-axis? Which quadrants can H lie in?

A line through parallel to the -axis has constant . So any point on it is of the form for any . If then is in Quadrant II, if in Quadrant III, if on the -axis. Thus can be in Quadrant II or III or on the axis.

3. Consider the points R (3, 0), A (0, –2), M (–5, –2) and P (–5, 2). If they are joined in the same order, predict: (i) Two sides of RAMP that are perpendicular to each other.

Plot shows on -axis, , horizontal, vertical. is horizontal and is vertical , so at . Also ? Check: Also is slanted. The clearly perpendicular pair is and (and also and the line through ).

3. Consider the points R (3, 0), A (0, –2), M (–5, –2) and P (–5, 2). If they are joined in the same order, predict: (ii) One side of RAMP that is parallel to one of the axes.

joins to , both have , so is parallel to the -axis. Also joins to , both have , so is parallel to the -axis.

3. Consider the points R (3, 0), A (0, –2), M (–5, –2) and P (–5, 2). If they are joined in the same order, predict: (iii) Two points that are mirror images of each other in one axis. Which axis will this be? Now plot the points and verify your predictions.

and have same and opposite , so they are mirror images in the -axis. Also and a reflection would be across . Plotting confirms and are symmetric with respect to the -axis.

4. Plot point Z (5, –6) on the Cartesian plane. Construct a right-angled triangle IZN and find the lengths of the three sides. (Comment: Answers may differ from person to person.)

Choose a convenient right angle, e.g. take vertically above and at the origin, then is vertical length , horizontal length , and . Any right triangle with one leg parallel to -axis and one to -axis will give legs as absolute differences of coordinates and hypotenuse via . Different choices of give different numbers but the method is the same.

5. What would a system of coordinates be like if we did not have negative numbers? Would this system allow us to locate all the points on a 2-D plane?

Without negative numbers we could only represent the first quadrant where and . Points to the left of the -axis or below the -axis would have no representation. So we could not locate points in Quadrants II, III, IV or on the negative axes. The full plane needs zero and negative numbers for the other three quadrants.

6. Are the points M (–3, –4), A (0, 0) and G (6, 8) on the same straight line? Suggest a method to check this without plotting and joining the points.

Check if distances satisfy collinearity: if lies between and , then . Compute , , . Since , the three points are collinear and is between and . Equivalently check slopes: and , equal so same line.

7. Use your method (from Problem 6) to check if the points R (–5, –1), B (–2, –5) and C (4, –12) are on the same straight line. Now plot both sets of points and check your answers.

Compute distances: , , . within rounding, but check exactly: , so not exactly equal, but close due to? Check slopes: slope , slope , so not collinear. Plotting shows a slight bend at .

8. Using the origin as one vertex, plot the vertices of: (i) A right-angled isosceles triangle.

Take , on -axis, on -axis. Then , angle at is , so triangle is right-angled and isosceles with legs and hypotenuse . Any equal legs on the axes work.

8. Using the origin as one vertex, plot the vertices of: (ii) An isosceles triangle with one vertex in Quadrant III and the other in Quadrant IV.

Keep . Choose in Quadrant III and in Quadrant IV. Then by , base . So triangle is isosceles with vertex at origin and base parallel to -axis.

9. The following table shows the coordinates of points S, M and T. In each case, state whether M is the midpoint of segment ST. Justify your answer. S (–3, 0) M (0, 0) T (3, 0)

Yes. Midpoint of and is . Also distances and .

9. The following table shows the coordinates of points S, M and T. In each case, state whether M is the midpoint of segment ST. Justify your answer. S (2, 3) M (3, 4) T (4, 5)

Yes. Midpoint of and is .

9. The following table shows the coordinates of points S, M and T. In each case, state whether M is the midpoint of segment ST. Justify your answer. S (0, 0) M (0, 5) T (0, –10)

No. Midpoint of and is . but , so is not the midpoint.

9. The following table shows the coordinates of points S, M and T. In each case, state whether M is the midpoint of segment ST. Justify your answer. S (–8, 7) M (0, –2) T (6, –3)

No. Midpoint of and is . Distances are unequal.

9. When M is the mid-point of ST, can you find any connection between the coordinates of M, S and T?

When is the midpoint, its coordinates are the average of the endpoints: if and then . So and .

10. Use the connection you found to find the coordinates of B given that M (–7, 1) is the midpoint of A (3, –4) and B (x, y).

Let . Then gives . And gives . So .

11. Let P, Q be points of trisection of AB, with P closer to A, and Q closer to B. Using your knowledge of how to find the coordinates of the midpoint of a segment, how would you find the coordinates of P and Q? Do this for the case when the points are A (4, 7) and B (16, –2).

Trisection divides into three equal parts. can be found as the midpoint of and , and as midpoint of and , or directly by section formula: and . For : , . Check: .

12. (i) Given the points A (1, –8), B (–4, 7) and C (–7, –4), show that they lie on a circle K whose center is the origin O (0, 0). What is the radius of circle K?

Distance from origin: , , . All three are equal to , so they lie on the circle centred at with radius units.

12. (ii) Given the points D (–5, 6) and E (0, 9), check whether D and E lie within the circle, on the circle, or outside the circle K.

, so lies inside . , so lies outside .

13. The midpoints of the sides of triangle ABC are the points D, E, and F. Given that the coordinates of D, E, and F are (5, 1), (6, 5), and (0, 3), respectively, find the coordinates of A, B and C.

Let be midpoint of , of , of . Using midpoint averages, solving gives vertices up to labelling permutation. Check: midpoint of and is , and is , and is . Any permutation of labels satisfying the same midpoint set is acceptable.

14. A city has two main roads which cross each other at the centre of the city. These two roads are along the North–South (N–S) direction and East–West (E–W) direction. All the other streets of the city run parallel to these roads and are 200 m apart. There are 10 streets in each direction. (i) Using 1 cm = 200 m, draw a model of the city in your notebook. Represent the roads/streets by single lines.

Draw a grid with the central cross as the axes. With cm = m, the city extends streets to each side of the centre. The grid lines are at cm and similarly for if centred, representing streets m apart. Represent each street by a single line; intersections are grid points.

14. A city has two main roads which cross each other at the centre of the city. These two roads are along the North–South (N–S) direction and East–West (E–W) direction. All the other streets of the city run parallel to these roads and are 200 m apart. There are 10 streets in each direction. (ii) (a) how many street intersections can be referred to as (4, 3).

The name refers to the crossing of the 4th N–S street and the 3rd E–W street. Counting from one edge, there is exactly one such crossing. If streets are numbered from any edge, there is one intersection with that label.

14. A city has two main roads which cross each other at the centre of the city. These two roads are along the North–South (N–S) direction and East–West (E–W) direction. All the other streets of the city run parallel to these roads and are 200 m apart. There are 10 streets in each direction. (ii) (b) how many street intersections can be referred to as (3, 4).

Similarly exactly one intersection is the 3rd N–S street with the 4th E–W street, namely point . It is a different point from , showing that order matters in coordinates.

15. A computer graphics program displays images on a rectangular screen whose coordinate system has the origin at the bottom-left corner. The screen is 800 pixels wide and 600 pixels high. A circular icon of radius 80 pixels is drawn with its centre at the point A (100, 150). Another circular icon of radius 100 pixels is drawn with its centre at the point B (250, 230). Determine: (i) whether any part of either circle lies outside the screen.

Screen covers , . For : left extent , right , bottom , top , so fully inside. For : extents , , all inside. So neither circle goes outside the screen.

15. A computer graphics program displays images on a rectangular screen whose coordinate system has the origin at the bottom-left corner. The screen is 800 pixels wide and 600 pixels high. A circular icon of radius 80 pixels is drawn with its centre at the point A (100, 150). Another circular icon of radius 100 pixels is drawn with its centre at the point B (250, 230). Determine: (ii) whether the two circles intersect each other.

Distance pixels. Sum of radii . Since , the circles overlap and intersect at two points. Since , one is not inside the other.

16. Plot the points A (2, 1), B (–1, 2), C (–2, –1), and D (1, –2) in the coordinate plane. Is ABCD a square? Can you explain why? What is the area of this square?

Compute sides: , , , . All sides equal. Diagonals: , , equal and bisect each other at origin. Also dot product of adjacent sides is zero, so angles are . Hence is a square centred at origin, rotated. Area = side square units. Also area = .