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1B_Ch8(1) 8.1 Rectangular Coordinates 1B_Ch8(2) A Introduction to Coordinate Systems B Rectangular Coordinate System Index 8.2 Distances and Areas in the 1B_Ch8(3) Rectangular Coordinate System A Distance between Two Points on a Horizontal or Vertical Line B Area of a Plane Figure Index 8.3 Polar Coordinates 1B_Ch8(4) A Introduction to Polar Coordinates B Comparison between Rectangular and Polar Coordinates Index 8.1 Rectangular Coordinates 1B_Ch8(5) Example A) Introduction to Coordinate Systems ‧ Refer to the following figure. A building is located in the area. We use D3 to represent its position. This kind of method for representing positions is called a coordinate system. Index 8.1 Index 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 Lily 4 3 James 2 Ann 1 A B C D E The position of Lily is B4 and the position of James is E3. Key Concept 8.1.1 Index 8.1 Rectangular Coordinates 1B_Ch8(7) B) Rectangular Coordinate System 1. Ordered Pairs ‧ An ordered pair is a pair of numbers written within brackets in a particular order. E.g. (1, 2), (7, –5) Index 8.1 Rectangular Coordinates 1B_Ch8(8) B) Rectangular Coordinate System 2. In a rectangular coordinate plane, we can locate the position of a point by its distances from the horizontal x-axis and vertical y-axis. Its position can be written as an ordered pair (a, b). y 3. In the figure, the ordered pair (a, b) denotes the coordinates of P where a is a b called the x-coordinate, b is called the y-coordinate. P(a, b) b O a x Index 8.1 Rectangular Coordinates 1B_Ch8(9) Example B) Rectangular Coordinate System y b O P(a, b) a x 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 8.1 Index 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 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) Fulfill Exercise Objective (a) The coordinates of P are (3, 1). (b) The coordinates of P are (–0.7, 7). Write down the coordinates of given points in a rectangular coordinate plane. Index 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 8.1 Rectangular Coordinates 1B_Ch8(13) Back to Question (a), (b) Fulfill Exercise Objective Find the coordinates of the point of intersection. From the graph, the required coordinates are (–4, 5). Key Concept 8.1.2 Index 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(14) A) Distance between Two Points on a Horizontal or Vertical Line (a) Any two points on the same y AB = x2 – x1 horizontal line have the same A y-coordinate. If A(x1, y) and B(x2, y) are these two points and x2 > x1, O B x then AB = x2 – x1. Index 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(15) Example A) Distance between Two Points on a Horizontal or Vertical Line (b) Any two points on the same vertical y P line have the same x-coordinate. PQ = y2 – y1 If P(x, y1) and Q(x, y2) are these two points and y2 > y1, Q O x then PQ = y2 – y1. Index 8.2 Index 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(16) Find the lengths of the line segments shown in the diagram. y M(–11, 6) N(2, 6) T(–7, 3) O x S(–7, –3) MN = [2 – (–11)] units = 13 units TS = [3 – (–3)] units = 6 units Index 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 Fulfill Exercise Objective Find the lengths of the sides or the perimeters of figures. = 240 units Index 8.2 Distances and Areas in the Rectangular Coordinate System In the figure, if AB = 7 units, 1B_Ch8(18) y B(b, 3) A(4, 3) find the value of b. O x Since A and B have the same y-coordinate (i.e. 3), AB is horizontal. Fulfill Exercise Objective From the figure, 4 > b ∴4–b =7 b = –3 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. Key Concept 8.2.1 Index 8.2 Distances and Areas in the Rectangular Coordinate System B) 1B_Ch8(19) Area of a Plane Figure 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 Index 8.2 Index 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. y A(–5, 6) 6 5 4 3 2 1 0 –7 –6 –5 –4 –3 –2 –1–1 –2 B(–5, –2)–3 x 1 2 3 4 5 6 7 8 9 10 C(8, –2) Index 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(21) Back to Question Base = BC Height = AB = [8 – (–5)] units = [6 – (–2)] units = 13 units = 8 units Area of △ABC = 1 × BC × AB 2 1 = × 13 × 8 sq. units 2 = 52 sq. units Index 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). (a) Find the length of HK. Soln (b) Find the height of △HKG with respect to the base HK. (c) Calculate the area of △HKG. Soln Soln Index 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 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(24) Back to Question 1 × HK × GN 2 1 = × 4 × 5 sq. units 2 (c) Area of △ HKG = = 10 sq. units Fulfill Exercise Objective Find areas of simple figures. Index 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 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 1 × (AD + BC) × AE 2 1 = × (2 + 6) × 4 sq. units 2 ∴ Area of ABCE = = 16 sq. units Fulfill Exercise Objective Find areas of simple figures. Key Concept Index 8.2.2 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 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). 1 Area of △KLM = × KM × LP 2 1 = × 9 × 3 sq. units 2 = 13.5 sq. units Index 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(29) Back to Question 1 Area of △KMN = × KM × NP 2 1 = × 9 × 6 sq. units 2 = 27 sq. units ∴ Area of KLMN = area of △KLM + area of △KMN = (13.5 + 27) sq. units = 40.5 sq.units Fulfill Exercise Objective Find areas of composite figures by splitting figures. Index 8.2 Distances and Areas in the Rectangular Coordinate System Find the area of pentagon y R(–5, 6) RMSTU in the figure. P 6 5 4 3 2 1 0 –7 –6 –5 –4 –3 –2 –1–1 –2 S(–5, –2)–3 1B_Ch8(30) U(8, 6) M(0, 2) x 1 2 3 4 5 6 7 8 9 10 T(8, –2) Join RS so that the figure becomes a rectangle RSTU. Then draw line segment MP as shown. Index 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(31) Back to Question 1 Area of △RMS = × RS × MP 2 1 = × 8 × 5 sq. units 2 = 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 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 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(33) Back to Question 1 Area of trapezium APQB = × (AP + BQ) × AB 2 1 = × (4 + 6) × 11 sq. units 2 = 55 sq. units 1 Area of △OAP = × OA × AP 2 1 = × 3 × 4 sq. units 2 = 6 sq. units Index 8.2 Distances and Areas in the Rectangular Coordinate System 1B_Ch8(34) Back to Question 1 Area of △RBQ = × RB × BQ 2 1 = × 6 × 6 sq. units 2 = 18 sq. units ∴ Area of OPQR = area of trapezium APQB – area of △OAP – area of △RBQ = (55 – 6 – 18) sq. units = 31 sq. units Fulfill Exercise Objective Find areas of figures by subtraction. Key Concept 8.2.2 Index 8.3 Polar Coordinates A) 1B_Ch8(35) Introduction to Polar Coordinates 1. In the polar coordinate plane in the figure, the point P is r units from the pole O. The angle which is measured anticlockwise from the polar axis OX to OP is θ. We can locate the position of P by r and θ, expressed as the ordered pair (r, θ). Index 8.3 Polar Coordinates 1B_Ch8(36) Example A) Introduction to Polar Coordinates 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 8.3 Index 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) N M 8 3 85 O X O 105 X (a) The polar coordinates of M are (8, 85). (b) The polar coordinates of N are (3, 105). Index 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). The polar coordinates of C are (5, 240). Fulfill Exercise Objective Write down the polar coordinates of points. Key Concept 8.3.1 Index 8.3 Polar Coordinates 1B_Ch8(39) Example B) Comparison between Rectangular and Polar Coordinates 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 8.3 Index 8.3 Polar Coordinates 1B_Ch8(40) Which figures you will use if measuring the length of OA and AB? y 4 3 A A 2 1 –5 –4 –3 –2 –1 O –1 1 2 3 4 5 x –2 B –3 B –4 –5 Fig. I Fig. II 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