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  1. 1B_Ch8(1)

  2. A B Introduction to Coordinate Systems Rectangular Coordinate System 1B_Ch8(2) 8.1 Rectangular Coordinates Index

  3. A B Distance between Two Points on a Horizontal or Vertical Line Area of a Plane Figure 1B_Ch8(3) 8.2 Distances and Areas in the Rectangular Coordinate System Index

  4. A B Introduction toPolar Coordinates Comparison betweenRectangular and Polar Coordinates 1B_Ch8(4) 8.3 Polar Coordinates Index

  5. ‧ Refer to the following figure. A building is located in the area. 8.1 Rectangular Coordinates 1B_Ch8(5) • Example Introduction to Coordinate Systems A) We use D3 to represent its position. This kind of method for representing positions is called a coordinate system. Index • Index 8.1

  6. A B C D E 8.1 Rectangular Coordinates 1B_Ch8(6) The figure shows the seating plan of Class 1A. It is known that the position of Ann is D2, indicate the position of Lily and James. 5 4 3 2 1 Lily James Ann The position of Lily is B4 and the position of James is E3. • Key Concept 8.1.1 Index

  7. 8.1 Rectangular Coordinates 1B_Ch8(7) Rectangular Coordinate System B) • Ordered Pairs ‧ An ordered pair is a pair of numbers written within brackets in a particular order. E.g. (1, 2), (7, –5) Index

  8. y a P(a, b) b b x a O 8.1 Rectangular Coordinates 1B_Ch8(8) Rectangular Coordinate System B) 2. In a rectangular coordinate plane, we can locate the position of a point by its distances from the horizontal x-axis and verticaly-axis. Its position can be written as an ordered pair (a, b). 3. In the figure, the ordered pair (a, b) denotes the coordinates of P where a is called the x-coordinate, b is called the y-coordinate. Index

  9. y P(a, b) b x O a 8.1 Rectangular Coordinates 1B_Ch8(9) • Example Rectangular Coordinate System B) 4. The intersection O(0, 0) of the x-axis and the y-axis is called the origin, which is the reference point of all points in the plane. Index • Index 8.1

  10. 8.1 Rectangular Coordinates 1B_Ch8(10) What are the coordinates of the origin O and the point C in the figure? The coordinates of O are (0, 0). The coordinates of C are (–4, –3). Index

  11. Fulfill Exercise Objective • Write down the coordinates of given points in a rectangular coordinate plane. 8.1 Rectangular Coordinates 1B_Ch8(11) Write down the coordinates of the point P in each of the following rectangular coordinate plane. (a) (b) (a) The coordinates of P are (3, 1). (b) The coordinates of P are (–0.7, 7). Index

  12. 8.1 Rectangular Coordinates 1B_Ch8(12) (a) Mark the four points A(–6, 7), B(1, 0), C(–13, –4) and D(2, 11) in the rectangular coordinate plane. (b) Draw a line through A and B and another line through C and D. What are the coordinates of the point of intersection? Index

  13. Fulfill Exercise Objective • Find the coordinates of the point of intersection. 8.1 Rectangular Coordinates 1B_Ch8(13) • Back to Question (a), (b) From the graph, the required coordinates are (–4, 5). • Key Concept 8.1.2 Index

  14. y AB = x2– x1 A B x O 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(14) Distance between Two Points on a Horizontal or Vertical Line A) (a) Any two points on the same horizontal line have the samey-coordinate. If A(x1, y) and B(x2, y) are these two points and x2 > x1,then AB = x2 – x1. Index

  15. y P PQ = y2– y1 Q x O 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(15) • Example Distance between Two Points on a Horizontal or Vertical Line A) (b) Any two points on the same vertical line have the same x-coordinate. If P(x, y1) and Q(x, y2) are these two points and y2 > y1, then PQ = y2 – y1. Index • Index 8.2

  16. y M(–11, 6) N(2, 6) T(–7, 3) x O S(–7, –3) 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(16) Find the lengths of the line segments shown in the diagram. = [2 – (–11)] units = 13 units = [3 – (–3)] units = 6 units MN TS Index

  17. Fulfill Exercise Objective • Find the lengths of the sides or the perimeters of figures. 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(17) A(–15, 30), B(–15, –20), C(55, –20) and D(55, 30) are four points in a rectangular coordinate plane. Given that ABCD is a rectangle, what is its perimeter? AB = [30 – (–20)] units = 50 units BC = [55 – (–15)] units = 70 units ∴ Perimeter of ABCD = (AB + BC) × 2 = (50 + 70) × 2 units = 240 units Index

  18. y B(b, 3) A(4, 3) x O Fulfill Exercise Objective • Given the distance between two points on the same horizontal or vertical line, find the coordinates or the unknown in the coordinates of a certain point. 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(18) In the figure, if AB = 7 units, find the value of b. Since A and B have the same y-coordinate (i.e. 3), AB is horizontal. From the figure, 4 > b ∴ 4 – b = 7 b = –3 • Key Concept 8.2.1 Index

  19. 1. We can find the areas of geometric figures in a rectangular coordinate plane by finding the lengths of some suitable vertical or horizontal line segments. • Example 2. Sometimes, indirect methods such as splitting or combining figures may be needed. • Example 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(19) Area of a Plane Figure B) Index • Index 8.2

  20. y A(–5, 6) 6 5 4 3 2 1 –1 –2 –3 x 0 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10 B(–5, –2) C(8, –2) 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(20) The figure shows a triangle with vertices at A(–5, 6), B(–5, –2) and C(8, –2). Calculate the area of △ABC. Index

  21. = ×BC×AB = × 13 × 8 sq. units 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(21) • Back to Question Base = BC = [8 – (–5)] units = 13 units Height = AB = [6 – (–2)] units = 8 units Area of △ABC = 52 sq. units Index

  22. 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(22) The figure shows a triangle with vertices at H(–1, –2), K(–1, 2) and G(4, 3). • Find the length of HK. • Find the height of △HKG with respect to the base HK. • Calculate the area of △HKG. • Soln • Soln • Soln Index

  23. 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(23) • Back to Question (a) HK = [2 – (–2)] units = 4 units (b)Through G, construct a perpendicular to HK to meet HK produced at N. The coordinates of N are (–1, 3). When HK is the base, height = GN = [4 – (–1)] units = 5 units Index

  24. = ×HK×GN = × 4 × 5 sq. units Fulfill Exercise Objective • Find areas of simple figures. 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(24) • Back to Question (c)Area of △HKG = 10 sq. units Index

  25. 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(25) In the figure, A(1, 2), B(–2, –2), C(4, –2) and D(3, 2) are the four vertices of a trapezium. Find the area of ABCD. AD = (3 – 1) units = 2 units BC = [4 – (–2)] units = 6 units Index

  26. = × (AD + BC)×AE = × (2 + 6) × 4 sq. units Fulfill Exercise Objective • Find areas of simple figures. 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(26) • Back to Question Through A, construct a perpendicular AE to BC. The coordinates of E are (1 , –2). AE = [2 – (–2)] units = 4 units ∴Area of ABCE = 16 sq. units • Key Concept 8.2.2 Index

  27. 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(27) In the figure, the vertices of the quadrilateral are K(–2, 5), L(–5, –3), M(–2, –4) and N(4, –3). Find the area of the quadrilateral. Index

  28. = ×KM×LP = × 9 × 3 sq. units 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(28) • Back to Question Join KM so that the figure is split into △KLM and △KMN. Then draw line segments LP and NP as shown. From the figure, the coordinates of P are (–2, –3). Area of △KLM = 13.5 sq. units Index

  29. = ×KM×NP = × 9 × 6 sq. units Fulfill Exercise Objective • Find areas of composite figures by splitting figures. 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(29) • Back to Question Area of △KMN = 27 sq. units ∴ Area of KLMN = area of △KLM + area of △KMN = (13.5 + 27) sq. units = 40.5 sq.units Index

  30. y R(–5, 6) U(8, 6) 6 5 4 3 2 1 –1 –2 –3 P M(0, 2) x 0 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8 9 10 S(–5, –2) T(8, –2) 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(30) Find the area of pentagon RMSTU in the figure. Join RS so that the figure becomes a rectangle RSTU. Then draw line segment MP as shown. Index

  31. = ×RS×MP = × 8 × 5 sq. units 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(31) • Back to Question Area of △RMS = 20 sq. units ∴ Area of RMSTU = area of RSTU – area of △RMS = [(13 × 8) – 20] sq. units = (104 – 20) sq. units = 84 sq. units Index

  32. 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(32) Find the area of quadrilateral OPQR in the figure. Draw two perpendiculars PA and QB to the x-axis. Then APQB is a trapezium, where the coordinates of A are (–3, 0) and the coordinates of B are (8, 0). Index

  33. = × (AP + BQ) ×AB = × (4 + 6) × 11 sq. units = ×OA×AP = × 3 × 4 sq. units 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(33) • Back to Question Area of trapezium APQB = 55 sq. units Area of △OAP = 6 sq. units Index

  34. = ×RB×BQ = × 6 × 6 sq. units Fulfill Exercise Objective • Find areas of figures by subtraction. 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(34) • Back to Question Area of △RBQ = 18 sq. units ∴ Area of OPQR = area of trapezium APQB – area of △OAP – area of △RBQ = (55 – 6 – 18) sq. units = 31 sq. units • Key Concept 8.2.2 Index

  35. 8.3 Polar Coordinates 1B_Ch8(35) Introduction to Polar Coordinates A) 1. In the polar coordinate plane in the figure, the point P is r units from the poleO. The angle which is measured anticlockwise from the polar axisOX to OP is θ. We can locate the position of P by r and θ, expressed as the ordered pair (r, θ). Index

  36. 8.3 Polar Coordinates 1B_Ch8(36) • Example Introduction to Polar Coordinates A) 2. The ordered pair (r, θ) denotes the polar coordinates of P, where r is the radius vector and θ is the polar angle. Index • Index 8.3

  37. N M 8 3 105 85 X O X O 8.3 Polar Coordinates 1B_Ch8(37) Write down the polar coordinates of the points M and N in the given polar coordinate plane. (a) (b) (a)The polar coordinates of M are (8, 85). (b)The polar coordinates of N are (3, 105). Index

  38. Fulfill Exercise Objective • Write down the polar coordinates of points. 8.3 Polar Coordinates 1B_Ch8(38) Write down the polar coordinates of the points A, B and C in the given polar coordinate plane. The polar coordinates of A are (4, 40). The polar coordinates of B are (3, 140). • Key Concept 8.3.1 The polar coordinates of C are (5, 240). Index

  39. 8.3 Polar Coordinates 1B_Ch8(39) • Example Comparison between Rectangular and Polar Coordinates B) 3. It is easier to find the distance between any point and O in a polar coordinate plane than in a rectangular coordinate plane. However, it is often difficult to find the distance between two points on a vertical or horizontal line in a polar coordinate plane. Index • Index 8.3

  40. y 4 3 2 1 –1 –2 –3 –4 –5 A A x –5 –4 –3 –2 –1 O 1 2 3 4 5 B B Fig. I Fig. II 8.3 Polar Coordinates 1B_Ch8(40) Which figures you will use if measuring the length of OA and AB? We use Fig. I to measure the length of OA and use Fig. II to measure the length of AB. • Key Concept 8.3.2 Index