Class 9 › Mathematics › Ganita Manjari Orienting Yourself: The Use of Coordinates Class 9 Mathematics · Ganita Manjari · Chapter 1
Summary Questions & answers Practise this chapter → 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 \(x\)-axis and the vertical \(y\)-axis — meet at the origin \(O = (0,0)\), dividing the plane into four quadrants. Every point is named by an ordered pair \((x, y)\), where \(x\) is its perpendicular distance from the \(y\)-axis and \(y\) its distance from the \(x\)-axis. Signs of the coordinates fix the quadrant: \((+,+)\), \((-,+)\), \((-,-)\) and \((+,-)\) for quadrants I to IV. Points on the \(x\)-axis have the form \((x,0)\) and points on the \(y\)-axis the form \((0,y)\). 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: \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). The midpoint of a segment is found by averaging coordinates.
Origin
The point where the \(x\)-axis and \(y\)-axis intersect, having coordinates \((0,0)\).
Quadrant
One of the four regions into which the coordinate axes divide the Cartesian plane.
Abscissa
The \(x\)-coordinate of a point, its perpendicular distance from the \(y\)-axis.
Ordinate
The \(y\)-coordinate of a point, its perpendicular distance from the \(x\)-axis.
Distance formula
The length between \((x_1,y_1)\) and \((x_2,y_2)\), equal to \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\).
Midpoint
The point halfway along a segment, with coordinates \(\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)\).
Q1 What are the coordinates of the origin? Answer Q2 In which quadrant does the point \((-3, 5)\) lie? Answer Q3 What is the \(x\)-coordinate of every point on the \(y\)-axis? Answer Q4 Find the distance between \(A(3,4)\) and \(D(7,1)\). Answer Q5 When is the point \((x,y)\) the same as \((y,x)\)? Answer Q6 How do you find the distance between two points lying on a horizontal line? Answer Q7 What are the coordinates of the midpoint of \((-3,0)\) and \((3,0)\)? Answer Q8 Why can three collinear points not all be vertices of a right triangle on the plane? Answer Q9 Point \(M\) is the midpoint of \(A(3,-4)\) and \(B\); if \(M=(-7,1)\), find \(B\). Answer Q10 How can you check whether three points are collinear without plotting? Answer Q1 (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? Answer Q2 (ii) What are the coordinates of D1 ? Answer Q3 (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? Answer Q4 (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? Answer Q1 1. What are the standard widths for a room door? Look around your home and in school. Answer Q2 2. Are the doors in your school suitable for people in wheelchairs? Answer Q1 1. What is the x-coordinate of a point on the y-axis? Answer Q2 2. Is there a similar generalisation for a point on the x-axis? Answer Q3 3. Does point Q (y, x) ever coincide with point P (x, y)? Justify your answer. Answer Q4 4. If x ≠ y, then (x, y) ≠ (y, x); and (x, y) = (y, x) if and only if x = y. Is this claim true? Answer Q1 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? Answer Q2 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? Answer Q3 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? Answer Q4 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? Answer Q5 3. Look at Reiaan’s bathroom. (i) What are the coordinates of the four corners O, F, R, and P of the bathroom? Answer Q6 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. Answer Q7 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. Answer Q8 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. Answer Q9 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. Answer Q1 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? Answer Q2 2. Can these distances help you find the distance AD? Answer Q1 1. What has remained the same and what has changed with this reflection? Answer Q2 2. Would these observations be the same if ΔADM is reflected in the x-axis (instead of the y-axis)? Answer Q1 1. What are the x-coordinate and y-coordinate of the point of intersection of the two axes? Answer Q2 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? Answer Q3 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. Answer Q4 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. Answer Q5 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. Answer Q6 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.) Answer Q7 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? Answer Q8 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. Answer Q9 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. Answer Q10 8. Using the origin as one vertex, plot the vertices of: (i) A right-angled isosceles triangle. Answer Q11 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. Answer Q12 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) Answer Q13 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) Answer Q14 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) Answer Q15 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) Answer Q16 9. When M is the mid-point of ST, can you find any connection between the coordinates of M, S and T? Answer Q17 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). Answer Q18 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). Answer Q19 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? Answer Q20 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. Answer Q21 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. Answer Q22 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. Answer Q23 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). Answer Q24 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). Answer Q25 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. Answer Q26 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. Answer Q27 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? Answer
What are the coordinates of the origin?
A \((1, 1)\) B \((0, 0)\) C \((1, 0)\) D \((0, 1)\)
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