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edmet-nklpunya.blogspot.com My Additional Mathematics Modules Form 4 (Version 2012) Topic 6: Mastering Coordinates Geometry by NgKL (M.Ed.,B.Sc.Hons.,Dip.Ed.,Dip.Edu.Mgt.,Cert.NPQH) edmet-nklpunya.blogspot.com edmet-nklpunya.blogspot.com COORDINATES GEOMETRY (GEOMETRI KOORDINAT) IMPORTANT NOTES (1) edmet-nklpunya.blogspot.com Exercise 6.1 – Distance Between Two Points 1. Find the distance between the following pairs of points. (a) A (6, 4) and B (1, -8) (b) A (6, 8) and B (3, 4) (c) C (9, -2) and D (3, 6) (d) P (7, -4) and Q (-5, 1) 2. Find the possible values of k for each of the following situation. (a) A (k, 4k), B (-k, 5k) and AB 80 unit (b) A (3, 1), B (6, k) and AB = 5 unit (c) A (2, 3), B (k, 5) and AB 20 unit (d) A (-2, 1), B (2, k) and AB = 5 unit edmet-nklpunya.blogspot.com Exercise 6.2 – Division of a Line Segment 1. Find the coordinate of the mid-point for straight lines of the following pairs of points: (a) A (3, -2) and B (-7, -6) (b) P (-2, 7) and Q (4, -3) (c) T (6, 4) and U (-2, 2) (d) V (-2, -3) and W (-6, -7) 2. Given that M is the mid point for the straight line AB. Find the coordinate of point B. (a) A (-1, 2) dan M (-4, 5) (b) A (2, 3) dan M (5, 4) (c) A (-2, 4) dan M (1, 5) (d) A (-6, 3) dan M (-2, -2) edmet-nklpunya.blogspot.com 3. ABCD is a parallelogram. Find the values of nilai h and k. (a) A(2, 3), B(5, 4), C(6, 7) and D(h, k) (b) A(-1,2), B(3, 5), C(h, k) dan D(-3, 7) 4. Find the coordinate of point P which divides the straight line AB to the ratio AP : PB as shown. (a) A(3, 5), B(3, -2) and AP : PB = 2 : 5 (b) A(-2, -2), B(3, 3) and AP : PB = 4 : 1 (c) A(-1, 5), B(4, 5) and AP : PB = 2 : 3 (d) A(-6, -9), B(2, 3) and AP : PB = 5 : 3 edmet-nklpunya.blogspot.com 5. Solve the following problems. (a) Given A(m, 7), B(1, n) and line AB is divided by point P(−5, 3) according to the ratio AP : PB = 2 : 3. Find the values of m and n. (b) Given P(x, 5), Q(−8, y) and line PQ is divided by point M(1 ,7) according to the ratio PM : MQ = 1 : 3. Find the values of x and y. (c) Given K( p, 4 ), L(3, n ) and line KL is divided by point M( 2, 6 ) according to the ratio KM : 2ML. Find the values of p and q. edmet-nklpunya.blogspot.com Exercise 6.3 Areas of Polygons (Triangles and Quadrilaterals) 1. Find the area of each of the following polygons with the given vertices. (a) A(-2, 4), B(0, -6) C(8, 4) (b) A(7, 0), B(1, 2), C(2, -5) (c) A(2, 5), B(6, 0), C(1, -3), D(-2, 2) (d) A(1, 6), B(1, 2), C(4, 1), D(6, 4) 2. Find the values of the unknown of a point with which the area of the triangle formed is given. (a) K(-2, −3), L(5, −1) M(p, 5), (b) A(3, h), B(2, 7), C(9, 10), Area of ∆KLM = 18 units2. Area of ∆ABC = 8.5 unit2. edmet-nklpunya.blogspot.com 3. Determine whether the following sets of points is collinear. (a) A(2, 1), B(14, 4) C(30, 8) (b) A(3, 5), B(−2, 6), C(5, −4) 4. Find the value of the unknown of a point which formed the sets of collinear points (a) A(5, w), B(1, −1) C(3, 3) (b) A(4, 0), B(0, y), C(2, 3) (c) A(12, 5), B(r, 3), C(−12, −1) (d) A(9, 6), B(3, s), C(4, 1), D(5, 4) edmet-nklpunya.blogspot.com COORDINATE GEOMETRY (GEOMETRI KOORDINAT) IMPORTANT NOTES (2) edmet-nklpunya.blogspot.com Exercise 6.4 – Gradient and Equations of Straight Lines 1. Find the gradient of the following pairs of points. Hence, find the equations of the straight lines. (a) (2, 6) and (8, 9) (b) (-2, 2) and (6, -2) (c) (-1, 3) and (-3, -1) (d) (3, -4) and (1, 0) 2. Diagrams show the x-intercept and y-intercept of each of the straight lines. Find each of the gradients by using the intercepts, hence find the equation of the straight lines. (a) (b) y y x −7 0 5 −5 x 0 4 edmet-nklpunya.blogspot.com 3. Given are pairs of the points which indicate the x-intercept and the y-intercept. Form the equation of each of the straight lines in intercept form. (a) A (3, 0) and B (0, −6) (b) P (−2, 0) and Q (0, −3) (c) T (6, 0) and U (0, 2) (d) V (−2, 0) and W (0, 7) 4. Given are the equations of straight lines in their intercept form. State the x-intercept, y-intercept and find the gradient of each of the straight lines. Equation in x-intercept y-intercept gradient intercept form x y a. 1 3 5 x y b. 1 5 7 x y c. 1 2 4 5. Find the equation of the straight line which has the gradient, m and passes through a point P as shown below. (a) m = 4, P (-7, -6) (b) m = −3, P (4, -3) (c) m = ⅜,, P (6, 4) (d) m = 2, P (−6, −7) edmet-nklpunya.blogspot.com 6. Given are the equations of straight lines in their gradient form. State the gradient, y-intercept,and x-intercept of each of the straight lines. Equation in x-intercept y-intercept gradient intercept form a. y = 7x + 3 b. y = −3x + 5 7. Transform the given straight line equations to the respective forms as indicated in the table below. Equation in intercept form Equation in gradient form Equation in general form a. y = −3x + 7 b. 2x + 5y – 4 = 0 x y c. 1 2 4 8. Point of Intersection of a Pair of Straight Lines – Solve by Simultaneous Equations Method. Given are a pair of equations of straight lines. Find the coordinate when the straight lines intersect each other. (a) 2x + 3y = 21 (b) x – 2y = 1 x+ y=8 4x + y + 5 = 0 edmet-nklpunya.blogspot.com Exercise 6.5 Parallel and Perpendicular Lines 1. Determine whether each of the following pair of straight lines are parallel or perpendicular to each other. x y (a) 2y + 3x = 12; 2 3 6 (b) 2x + y =2; 4x – 2y = 3 x y (c) 3y + x – 5 = 0 ; y = 3x + 1 (d) 3 6 1 ; 2y + 4x = 3 2. Solve the following problems. (a) Find the equation of the straight line which passes through point P(−1, 3), and is parallel to 3x – 2y = 6. Write the equation in general form. x y (b) Find the equation of the straight line which passes through point P(4, −3), and is parallel to 1 . 3 6 Write the equation in gradient form. edmet-nklpunya.blogspot.com (c) Find the equation of the straight line which passes through point P(−2, 4), and is perpendicular to 3x + y = 6. Write the equation in general form. (d) Find the equation of the straight line which passes through point P(0, −6 ), and is perpendicular to 2x – y +3 = 0. Write the equation in gradient form. Exercise 6.6 – Equation of Locus 1. Determine the equation of the locus of the moving point P(x, y) that satisfies the following condition. (a) Point P(x, y) moves such that it is 3 units from (b) Point P(x, y) moves such that it is 5 units from point Q (1, 5). point R(−2, 5). (c) Point P(x, y) moves such that its distance from (d) Point P(x, y) moves such that it is equidistant from point A(4,8) is twice its distance from point B(0, point M(2, 4) and N(−3, 12). 3). edmet-nklpunya.blogspot.com (e) Point P(x, y) moves such that its distance from (d) Point P(x, y) moves such that its distance from point K(5,0) and point L(1, −4) is in the ratio point R(2, 4) is 6 units and point S(−3, 12) is 4 PK : PL = 1 : 3 . unit. Exercise 6.7 – Past Year SPM Questions 1. SPM2009/Paper 1/Q15 y Diagram 15 shows a straight line AC. A(−2, 3) The point B lies on AC such that AB : BC = 3 : 1 Find the coordinates of B. B(h, k) x O C(14, 0) Diagram 15 [3 marks] 2. SPM2008/Paper 1/Q13. Diagram 13 shows a straight line passing through S(3, 0) and T (0, 4). (a) Write down the equation of the straight line y x y in the form 1. T(0, 4) a b (b) A point P(x, y) moves such that PS = PT. Find the equation of the locus of P. x O S(3, 0) Diagram 13 [4 marks] edmet-nklpunya.blogspot.com 3. SPM2009/Paper 2/Q9 Solution by scale drawing is not accepted. Diagram 9 shows a trapezium OABC. The line OA is perpendicular to line AB, which intersects the y-axis at 3 point Q. It is given that equation of OA is y = - x and the equation of AB is 6y = kx + 26. 2 y (a) Find B (i) the value of k. 6y= kx + 26 Q A 3 y = x C 2 O x (ii) the coordinate of A [4 marks] Diagram 9 (b) Given AQ : QB = 1 : 2, find (i) the coordinate of B. (ii) the equation of the straight line BC. [4 marks] (c) A point P(x, y) moves such that 2PA = PB. Find the equation of the locus of P. [2 marks] edmet-nklpunya.blogspot.com 4. SPM2008/Paper 2/Q10 3 Solution by scale drawing is not accepted. y P( , k) 2 Diagram 10 shows a triangle OPQ. Point S lies on the line PQ. (a) A point W moves such that its distance S(3, 1) from point S is always 2½ units. Find the equation of the locus of W. [3 marks] O x Q Diagram 10 (b) It is given that point P and point Q lie on the locus of W. Calculate (i) the value of k. (ii) the coordinate of Q. [5 marks] (c) Hence, find the area, in unit2, of triangle OPQ. [2 marks] edmet-nklpunya.blogspot.com 5. SPM2007/Paper 1/Q13 y The straight line x 1 has a y-intercept of 2 and parallel to the straight line y = kx = 0. Determine the 6 h value of h and of k. [3 marks] 6. SPM2007/Paper 2/Q2 Solution by scale drawing will not be accepted. In Diagram 1, the straight line AB has an equation y y + 2x + 8 = 0 . AB intersects the x-axis at point A intersects the y-axis at point B. A x Point P lies on AB such that AP : PB = 1 : 3. P Find (a) the coordinates of P. [3 marks] B y + 2x + 8 = 0 Diagram 1 (b) the equation of the straight line that passes through P and is perpendicular to AB. [3 marks] edmet-nklpunya.blogspot.com 7. SPM2010/Paper 1/Q13 The straight line passes through A(−2, −5) and B(6, 7). (a) Given C (h,10) lies on the straight line AB. Find the value of h. (b) Point D divides the lines segment AB in the ratio 1: 3, Find the coordinates of D. [4 marks] 8. SPM2010/Paper 2/Q5 Solution by scale drawing will not be accepted. Diagram 5 shows the straight line AC which intersect the y-axis at point B. The equation of AC is 3y = 2x −15 . Find (a) the equation of the straight line which passes through point A and is perpendicular to AC [4 marks] y C 0 x B (b) (i) the coordinates of B. Diagram 5 A(−3, −7) (ii) the coordinates of C, given AB : BC = 2 : 7 [3 marks] The End

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